On the ampleness of the cotangent bundles of complete intersections
OON THE AMPLENESSOF THE COTANGENT BUNDLESOF COMPLETE INTERSECTIONS
SONG-YAN XIEA bstract . Based on a geometric interpretation of Brotbek’s symmetric di ff erential forms, for theintersection family X of generalized Fermat-type hypersurfaces in P N K defined over any field K ,we construct (cid:14) reconstruct explicit symmetric di ff erential forms by applying Cramer’s rule, skippingcohomology arguments, and we further exhibit unveiled families of lower degree symmetric di ff er-ential forms on all possible intersections of X with coordinate hyperplanes.Thereafter, we develop what we call the ‘ moving coe ffi cients method ’ to prove a conjecture madeby Olivier Debarre: for generic c (cid:62) N / hypersurfaces H , . . . , H c ⊂ P N C of degrees d , . . . , d c su ffi ciently large, the intersection X : = H ∩· · ·∩ H c has ample cotangent bundle Ω X , and concerninge ff ectiveness, the lower bound d , . . . , d c (cid:62) N N works.Lastly, thanks to known results about the Fujita Conjecture, we establish the very-ampleness of Sym κ Ω X for all κ (cid:62) (cid:16) (cid:80) ci = d i (cid:17) . Introduction
In 2005, Debarre established that, in a complex abelian variety of dimension N , for c (cid:62) N / ffi ciently ample generic hypersurfaces H , . . . , H c , their intersection X : = H ∩ · · · ∩ H c hasample cotangent bundle Ω X , thereby answering a question of Lazarsfeld (cf. [18]). Then naturally,by thoughtful analogies between geometry of Abelian varieties and geometry of projective spaces,Debarre proposed the following conjecture in Section 3 of [18], extending in fact an older questionraised by Schneider [58] in the surface case: Conjecture 1.1. [Debarre Ampleness Conjecture]
For all integers N (cid:62)
2, for every integer N / (cid:54) c < N , there exists a positive lower bound: d (cid:29) d , . . . , d c (cid:62) d , for generic choices of c hypersurfaces: H i ⊂ P N C ( i = ··· c ) with degrees: deg H i = d i , Mathematics Subject Classification.
Key words and phrases.
Complex hyperbolicity, Fujita Conjecture, Debarre Ampleness Conjecture, Generic,Complete intersection, Cotangent bundle, Cramer’s rule, Symmetric di ff erential form, Moving Coe ffi cients Method,Base loci, Fibre dimension estimate, Core Lemma, Gaussian elimination.This work was supported by the Fondation Mathématique Jacques Hadamard through the grant N o ANR-10-CAMP-0151-02 within the “Programme des Investissements d’Avenir”. a r X i v : . [ m a t h . AG ] J a n he intersection: X : = H ∩ · · · ∩ H c has ample cotangent bundle Ω X .Precisely, according to a ground conceptualization due to Hartshorne [36], the expected ample-ness is that, for all large degrees k (cid:62) k (cid:29)
1, the global symmetric k -di ff erentials on X : Γ (cid:0) X , Sym k Ω X (cid:1) are so abundant and diverse, that firstly, at every point x ∈ X , the first-order jet evaluation map: Γ (cid:0) X , Sym k Ω X (cid:1) (cid:16) Jet Sym k Ω X (cid:12)(cid:12)(cid:12) x is surjective, where for every vector bundle E → X the first-order jet of E at x is defined by: Jet E (cid:12)(cid:12)(cid:12) x : = O x ( E ) (cid:46) ( m x ) O x ( E ) , and that secondly, at every pair of distinct points x (cid:44) x in X , the simultaneous evaluation map: Γ (cid:0) X , Sym k Ω X (cid:1) (cid:16) Sym k Ω X (cid:12)(cid:12)(cid:12) x ⊕ Sym k Ω X (cid:12)(cid:12)(cid:12) x is also surjective.The hypothesis: c (cid:62) n appears optimal , for otherwise when c < n , there are no nonzero global sections for all degrees k (cid:62) Γ (cid:0) X , Sym k Ω X (cid:1) = , according to Brückmann-Rackwitz [8] and Schneider [58], whereas, in the threshold case c = n ,nonzero global sections are known to exist.As highlighted in [18], projective varieties X having ample cotangent bundles enjoy severalfascinating properties, for instance the following ones. • All subvarieties of X are all of general type. • There are finitely many nonconstant rational maps from any fixed projective variety to X ([51]). • If X is defined over C , then X is Kobayashi-hyperbolic, i.e. every holomorphic map C → X must be constant ([20, p. 16, Proposition 3.1], [27, p. 52, Proposition 4.2.1]). • If X is defined over a number field K , the set of K -rational points of X is expected to befinite (Lang’s conjecture, cf. [38], [48]).Since ampleness of cotangent bundles potentially bridges Analytic Geometry and ArithmeticGeometry in a deep way, it is interesting to ask examples of such projective varieties. In one-dimensional case, they are in fact our familar Riemann surfaces (cid:14) algebraic curves with genus (cid:62) X , . . . , X (cid:96) are smooth complex projective varietieshaving positive dimensions: dim X i (cid:62) d (cid:62) ( i = ··· (cid:96) ) , ll of whose Serre line bundles O P (T Xi ) (1) → P (T X i ) enjoy bigness: dim Γ (cid:0) P (T X i ) , O P (T Xi ) ( k ) (cid:1) = dim Γ (cid:0) X i , Sym k Ω X i (cid:1) (cid:62) k → ∞ constant (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) > · k X i − , then a generic complete intersection: Y ⊂ X × · · · × X (cid:96) having dimension: dim Y (cid:54) d ( (cid:96) + +
12 ( d + Ω Y .In his Ph.D. thesis under the direction of Mourougane, Brotbek [5] reached an elegant proof ofthe Debarre Ampleness Conjecture in dimension n =
2, in all codimensions c (cid:62)
2, for genericcomplete intersections X ⊂ P + c ( C ) having degrees: d , . . . , d c (cid:62) n + c ) + n + c − , by extending the techniques of Siu [62, 63, 64], Demailly [20, 23, 22], Rousseau [56], P˘aun [52,53], Merker [43], Diverio-Merker-Rousseau [26], Mourougane [50], and by employing the conceptof ampleness modulo a subvariety introduced by Miyaoka in [47]. Also, for smooth completeintersections X n ⊂ P n + c ( C ) with c (cid:62) n (cid:62)
2, Brotbek showed using holomorphic Morse inequalitiesthat when: d , . . . , d c (cid:62) (cid:20) n − (cid:0) n − (cid:1) n n + c + (cid:32) n − n (cid:33) + (cid:21)(cid:32) n (cid:98) n (cid:99) (cid:33) (2 n + c )!( n + c )! ( c − n )! c ! , bigness of the Serre line bundle O P (T X ) (1) → P (T X ) holds: dim Γ (cid:0) P (T X ) , O P (T X ) ( k ) (cid:1) = dim Γ (cid:0) X , Sym k Ω X (cid:1) (cid:62) k → ∞ χ Euler (cid:0) X , Sym k Ω X (cid:1) (cid:62) k → ∞ constant (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) > · k n − , whereas a desirable control of the base locus of the inexplicitly given nonzero holomorphic sectionsseems impossible by means of currently available techniques.To find an alternative approach, a key breakthrough happened in 2014, when Brotbek [7] ob-tained explicit global symmetric di ff erential forms in coordinates by an intensive cohomologicalapproach. More specifically, under the assumption that the ambient field K has characteristic zero,using exact sequences and the snake lemma, Brotbek firstly provided a key series of long injectivecohomology sequences, whose left initial ends consist of the most general global twisted sym-metric di ff erential forms, and whose right target ends consist of huge dimensional linear spaceswell understood. Secondly, Brotbek proved that the image of each left end, going through the fullinjections sequence, is exactly the kernel of a certain linear system at the right end. Thirdly, byfocusing on pure Fermat-type hypersurface equations ([7, p. 26]): F j = N (cid:88) i = s ji Z ei ( j = ··· c ) , (1)with integers c (cid:62) N / e (cid:62)
1, where s ji are some homogeneous polynomials of the same degree (cid:15) (cid:62)
0, Brotbek step-by-step traced back some kernel elements from each right end all the way to he left end, every middle step being an application of Cramer’s rule, and hence he constructedglobal twisted symmetric di ff erential forms with neat determinantal structures ([7, p. 27–31]).Thereafter, by employing the standard method of counting base-locus-dimension in two ways inalgebraic geometry (see e.g. Lemma 8.15 below), Brotbek established that the Debarre AmplenessConjecture holds when: 4 c (cid:62) N − , for equal degrees: d = · · · = d c (cid:62) N + , (2)the constructions being flexible enough to embrace ‘approximately equal degrees’, in the samesense as Theorem 5.2 below.Inspired much by Brotbek’s works, we propose the following answer to the Debarre AmplenessConjecture. Theorem 1.2.
The cotangent bundle of the intersection in P N C of at least N / generic hypersurfaceswith degrees (cid:62) N N is ample. In fact, we will prove the following main theorem, which coincides with the above theorem for r = K = C , and whose e ff ective bound d = N N will be obtained in Theorem 11.2. Theorem 1.3 ( Ampleness).
Over any field K which is not finite, for all positive integers N (cid:62) ,for any nonnegative integers c , r (cid:62) with: c + r (cid:62) N , there exists a lower bound d (cid:29) such that, for all positive integers:d , . . . , d c , d c + , . . . , d c + r (cid:62) d , for generic c + r hypersurfaces: H i ⊂ P N K ( i = ··· c + r ) with degrees: deg H i = d i , the cotangent bundle Ω V of the intersection of the first c hypersurfaces:V : = H ∩ · · · ∩ H c restricted to the intersection of all the c + r hypersurfaces:X : = H ∩ · · · ∩ H c ∩ H c + ∩ · · · ∩ H c + r is ample. First of all, remembering that ampleness (or not) is preserved under any base change obtainedby ambient field extension , one only needs to prove the Ampleness Theorem 1.3 for algebraicallyclosed fields K .Of course, we would like to have d = d ( N , c , r ) as small as possible, yet the optimal one is atpresent far beyond our reach, and we can only get exponential ones like: d = N N (Theorem 11 . , which confirms the large degree phenomena in Kobayashi hyperbolicity related problems ([26, 2,22, 16, 64]). When 2 (2 c + r ) (cid:62) N −
2, we obtain linear bounds for equal degrees: d = · · · = d c + r (cid:62) N + , ence we recover the lower bounds (2) in the case r =
0, and we also obtain quadratic bounds forall large degrees: d , . . . , d c + r (cid:62) (3 N + N + . Better estimates of the lower bound d will be explained in Section 12.Concerning the proof, primarily, as anticipated (cid:14) emphasized by Brotbek and Merker ([7, 44]), itis essentially based on constructing su ffi ciently many global negatively twisted symmetric di ff er-ential forms, and then inevitably, one has to struggle with the overwhelming di ffi culty of clearingout their base loci, which seems, at the best of our knowledge, to be an incredible mission.In order to bypass the complexity in these two aspects, the following seven ingredients areindispensable in our approach: (cid:192) generalized Brotbek’s symmetric di ff erential forms (Subsection 6.10); (cid:193) global moving coe ffi cients method (MCM) (Subsection 7.2); (cid:194) ‘hidden’ symmetric forms on intersections with coordinate hyperplanes (Subsection 6.4); (cid:195) MCM on intersections with coordinate hyperplanes (Subsection 7.3); (cid:196)
Algorithm of MCM (Subsection 7.1); (cid:197)
Core Lemma of MCM (Section 10); (cid:198) product coup (Subsection 5.3).In fact, (cid:192) is based on a geometric interpretation of Brotbek’s symmetric di ff erential forms ([7,Lemma 4.5]), and has the advantage of producing symmetric di ff erential forms by directly copyinghypersurface equations and their di ff erentials. Facilitated by (cid:193) , which is of certain combinatorialinterest, (cid:192) amazingly cooks a series of global negatively twisted symmetric di ff erential forms,which are of nice uniform structures. However, unfortunately, one still has the di ffi culty that allthese obtained global symmetric forms happen to coincide with each other on the intersections withany two coordinate hyperplanes, so that their base locus stably keeps positive (large) dimension,which is an annoying obstacle to ampleness.Then, to overcome this di ffi culty enters (cid:194) , which is arguably the most critical ingredient inharmony with MCM, and whose importance is much greater than its appearance as somehow acorollary of (cid:192) . Thus, to compensate the defect of (cid:192) - (cid:193) , it is natural to design (cid:195) which completesthe framework of MCM. And then, (cid:196) is smooth to be devised, and it provides suitable hypersur-face equations for MCM. Now, the last obstacle to amplness is about narrowing the base loci, anultimate di ffi culty solved by (cid:197) . Thus, the Debarre Conjecture is settled in the central cases of al-most equal degrees. Finally, the magical coup (cid:198) thereby embraces all large degrees for the DebarreConjecture, and naturally shapes the formulation of the Ampleness Theorem.Lastly, taking account of known results about the Fujita Conjecture in Complex Geometry (cf.survey [21]), we will prove in Section 13 the following Theorem 1.4 ( E ff ective Very Ampleness). Under the same assumption and notation as in theAmpleness Theorem 1.3, if in addition the ambient field K has characteristic zero, then for genericchoices of H , . . . , H c + r , the restricted cotangent bundle Sym κ Ω V (cid:12)(cid:12)(cid:12) X is very ample on X, for every κ (cid:62) κ , with the uniform lower bound: κ = (cid:16) c (cid:88) i = d i + c + r (cid:88) i = d i (cid:17) . n the end, we would like to propose the following Conjecture 1.5. (i)
Over an algebraically closed field K , for any smooth projective K -variety P with dimension N , for any integers c , r (cid:62) c + r (cid:62) N , for any very ample line bundles L , . . . , L c + r on P , there exists a lower bound: d = d ( P , L • ) (cid:29) d , . . . , d c , d c + , . . . , d c + r (cid:62) d , for generic choices of c + r hypersurfaces: H i ⊂ P ( i = ··· c + r ) defined by global sections: F i ∈ H (cid:0) P , L ⊗ d i i (cid:1) , the cotangent bundle Ω V of the intersection of the first c hypersurfaces: V : = H ∩ · · · ∩ H c restricted to the intersection of all the c + r hypersurfaces: X : = H ∩ · · · ∩ H c ∩ H c + ∩ · · · ∩ H c + r is ample. (ii) There exists a uniform lower bound: d = d ( P ) (cid:29) L • . (iii) There exists a uniform lower bound: κ = κ ( P ) (cid:29) d , . . . , d c + r , such that for generic choices of H , . . . , H c + r , the restricted cotangentbundle Sym κ Ω V (cid:12)(cid:12)(cid:12) X is very ample on X , for every κ (cid:62) κ . Acknowledgements.
First of all, I would like to express my sincere gratitude to my thesis advisorJoël Merker, for his suggestion of the Debarre Ampleness Conjecture, for his exemplary guidance,patience, encouragement, and notably for his invaluable instructions in writing and in L A TEX, aswell as for his prompt help in the introduction.Mainly, I deeply express my debt to the recent breakthrough [7] by Damian Brotbek. Especially,I thank him for explaining and sharing his ideas during a private seminar in Merker’s o ffi ce on16 October 2014; even beyond the Debarre conjecture, it became clear that Damian Brotbek’sapproach will give higher order jet di ff erentials on certain classes of hypersurfaces. Next, I wouldlike to thank Junjiro Noguchi for the interest he showed on the same occasion, when I presentedfor the first time the moving coe ffi cients method which led to this article.Moreover, I thank Zhi Jiang, Yang Cao, Yihang Zhu and Zhizhong Huang for helpful discus-sions. Also, I thank Lionel Darondeau, Junyan Cao, Huynh Dinh Tuan and Tongnuo Wei for theirencouragement.Lastly, I heartily thank Olivier Debarre for suggesting the title, and Nessim Sibony, JunjiroNoguchi, Damian Brotbek, Lie Fu, Yihang Zhu, Zhizhong Huang for useful remarks. . Preliminaries and Restatements of the Ampleness Theorem 1.3
Two families of hypersurface intersections in P N K . Fix an arbitrary algebraically closed field K . Now, we introduce the fundamental object of this paper: the intersection family of c + r hypersurfaces with degrees d , . . . , d c + r (cid:62) K -projective space P N K of dimension N , equippedwith homogeneous coordinates [ z : z : · · · : z N ].Recalling that the projective parameter space of degree d (cid:62) P N K is: P (cid:16) H (cid:0) P N K , O P N K ( d ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) dim K = ( N + dN ) (cid:17) = P (cid:110) (cid:88) | α | = d A α z α : A α ∈ K (cid:111) , we may denote by: P (cid:16) ⊕ c + ri = H (cid:0) P N K , O P N K ( d i ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) dim K = (cid:80) c + ri = ( N + diN ) (cid:17) = P (cid:110) ⊕ c + ri = (cid:88) | α | = d i A i α z α : A i α ∈ K (cid:111) the projective parameter space of c + r hypersurfaces with degrees d , . . . , d c + r . This K -projectivespace has dimension: ♦ : = c + r (cid:88) i = (cid:32) N + d i N (cid:33) − , (3)hence we write it as: P ♦ K = Proj K (cid:2) { A i α } (cid:54) i (cid:54) c + r | α | = d i (cid:3) , (4)where, as shown above, A i α are the homogeneous coordinates indexed by the serial number i ofeach hypersurface and by all multi-indices α with the weight | α | = d i associated to the degree d i monomials z α ∈ K [ z , . . . , z N ].Now, we introduce the two subschemes: X ⊂ V ⊂ P ♦ K × K P N K , where X is defined by ‘all’ the c + r bihomogeneous polynomials: X : = V (cid:16) (cid:88) | α | = d A α z α , . . . , (cid:88) | α | = d c A c α z α , (cid:88) | α | = d c + A c + α z α , . . . , (cid:88) | α | = d c + r A c + r α z α (cid:17) , (5)and where V is defined by the ‘first’ c bihomogeneous polynomials: V : = V (cid:16) (cid:88) | α | = d A α z α , . . . , (cid:88) | α | = d c A c α z α (cid:17) . (6)Then we view X , V ⊂ P ♦ K × K P N K as two families of closed subschemes of P N K parametrized bythe projective parameter space P ♦ K .2.2. The relative cotangent sheaves family of V . A comprehensive reference on sheaves ofrelative di ff erentials is [42, Section 6 . . , pr be the two canonical projections: P ♦ K × K P N K pr (cid:123) (cid:123) pr (cid:36) (cid:36) P ♦ K P N K . (7) hen, by composing with the subscheme inclusion: i : V (cid:44) → P ♦ K × K P N K , we receive a morphism: pr ◦ i : V −→ P ♦ K , together with a sheaf Ω V / P ♦ K of relative di ff erentials of degree 1 of V over P ♦ K .Since pr is of finite type and P ♦ K is noetherian, a standard theorem ([42, p. 216, Proposi-tion 1 . Ω V / P ♦ K is coherent.We may view Ω V / P ♦ K as the family of the cotangent bundles for the intersection family V , sincethe coherent sheaf Ω V / P ♦ K is indeed locally free on the Zariski open set that consists of smoothcomplete intersections.2.3. The projectivizations and the Serre line bundles.
We refer the reader to [37, pp. 160-162]for the considerations in this subsection.Starting with the noetherian scheme V and the coherent degree 1 relative di ff erential sheaf Ω V / P ♦ K on it, we consider the sheaf of relative O V -symmetric di ff erential algebras: Sym • Ω V / P ♦ K : = (cid:77) i (cid:62) Sym i Ω V / P ♦ K . According to the construction of [37, p. 160], noting that this sheaf has a natural structure ofgraded O V -algebras, and moreover that it satisfies the condition ( † ) there, we receive the projec-tivization of Ω V / P ♦ K : P (cid:0) Ω V / P ♦ K (cid:1) : = Proj (cid:0)
Sym • Ω V / P ♦ K (cid:1) . (8)As described in [37, p. 160], P ( Ω V / P ♦ K ) is naturally equipped with the so-called Serre line bundle O P ( Ω V / P ♦ K ) (1) on it.Similarly, replacing V by P ♦ K × K P N K , we obtain the relative di ff erentials sheaf of P ♦ K × K P N K withrespect to pr in (7): Ω P ♦ K × K P N K / P ♦ K (cid:27) pr ∗ Ω P N K , and we thus obtain its projectivization: P (cid:0) Ω P ♦ K × K P N K / P ♦ K (cid:1) : = Proj (cid:0)
Sym • Ω P ♦ K × K P N K / P ♦ K (cid:1) (cid:27) P ♦ K × K Proj (cid:0)
Sym • Ω P N K (cid:1) . (9)We will abbreviate Proj (cid:0)
Sym • Ω P N K (cid:1) as P ( Ω P N K ), and denote its Serre line bundle by O P ( Ω P N K ) (1).Then, the Serre line bundle O P ( Ω P ♦ K × KP N K / P ♦ K ) (1) on the left hand side of (9) is nothing but the linebundle (cid:101) π ∗ O P ( Ω P N K ) (1) on the right hand side, where (cid:101) π is the canonical projection: (cid:101) π : P ♦ K × K P ( Ω P N K ) → P ( Ω P N K ) . (10)Now, note that the commutative diagram: V (cid:31) (cid:127) i (cid:47) (cid:47) pr ◦ i (cid:35) (cid:35) P ♦ K × K P N K pr (cid:15) (cid:15) P ♦ K nduces the surjection (cf. [37, p. 176, Proposition 8.12]): i ∗ Ω P ♦ K × K P N K / P ♦ K (cid:16) Ω V / P ♦ K , and hence yields the surjection: i ∗ Sym • Ω P ♦ K × K P N K / P ♦ K (cid:16) Sym • Ω V / P ♦ K . Taking ‘
Proj ’, thanks to (9), we obtain the commutative diagram: P ( Ω V / P ♦ K ) (cid:15) (cid:15) (cid:31) (cid:127) (cid:101) i (cid:47) (cid:47) P ♦ K × K P ( Ω P N K ) (cid:15) (cid:15) V (cid:31) (cid:127) i (cid:47) (cid:47) P ♦ K × K P N K . (11)Thus, the Serre line bundle O P ( Ω V / P ♦ K ) (1) becomes exactly the pull back of ‘the Serre line bundle’ (cid:101) π ∗ O P ( Ω P N K ) (1) under the inclusion (cid:101) i : O P ( Ω V / P ♦ K ) (1) = (cid:101) i ∗ (cid:0)(cid:101) π ∗ O P ( Ω P N K ) (1) (cid:1) = ( (cid:101) π ◦ (cid:101) i ) ∗ O P ( Ω P N K ) (1) . (12)2.4. Restatement of Theorem 1.3.
Let (cid:101) π be the canonical projection: (cid:101) π : P ♦ K × K P ( Ω P N K ) → P ♦ K × K P N K , and let π , π be the compositions of (cid:101) π with pr , pr : P ♦ K × K P ( Ω P N K ) π : = pr ◦ (cid:101) π (cid:3) (cid:3) π : = pr ◦ (cid:101) π (cid:28) (cid:28) (cid:101) π (cid:15) (cid:15) P ♦ K × K P N K pr (cid:121) (cid:121) pr (cid:37) (cid:37) P ♦ K P N K . (13)Let: P : = (cid:101) π − ( X ) ∩ P ( Ω V / P ♦ K ) ⊂ P ( Ω V / P ♦ K ) ⊂ P ♦ K × K P ( Ω P N K ) (14)be ‘the pullback’ of: X ⊂ V ⊂ P ♦ K × K P N K under the map (cid:101) π , and let: O P (1) : = O P ( Ω V / P ♦ K ) (1) (cid:12)(cid:12)(cid:12) P = (cid:101) π ∗ O P ( Ω P N K ) (1) (cid:12)(cid:12)(cid:12) P [see (12) ] (15)be the restricted Serre line bundle.Now, we may view P as a family of subschemes of P ( Ω P N K ) parametrized by the projective pa-rameter space P ♦ K under the restricted map: π : P −→ P ♦ K . (16) hus Theorem 1.3 can be reformulated as below, with the assumption that the hypersurface degrees d , . . . , d c + r are su ffi ciently large: d , . . . , d c + r (cid:29) . Theorem 1.3 (Version A).
For a generic point t ∈ P ♦ K , over the fibre: P t : = π − ( t ) ∩ P , the restricted Serre line bundle: O P t (1) : = O P (1) (cid:12)(cid:12)(cid:12) P t (17) is ample. From now on, every closed point: t = (cid:104) { A i α } (cid:54) i (cid:54) c + r | α | = d (cid:105) ∈ P ♦ K will be abbreviated as: t = [ F : · · · : F c + r ] , where: F i : = (cid:88) | α | = d i A i α z α ( i = ··· c + r ) . Then we have: P t = { t } × K F c + ,..., F c + r P F ,..., F c , for a uniquely defined subscheme: F c + ,..., F c + r P F ,..., F c ⊂ P (cid:0) Ω P N K (cid:1) . (18) Theorem 1.3 (Version B).
For a generic closed point: [ F : · · · : F c + r ] ∈ P ♦ K , the Serre line bundle O P ( Ω P N K ) (1) is ample on F c + ,..., F c + r P F ,..., F c . To have a better understanding of the above statements, we now investigate the geometry behind.3.
The background geometry
Since K is an algebraically closed field, throughout this section, we view each scheme in theclassical sense (cf. [37, Chapter 1]), i.e. its underlying topological space ( K -variety) consists of allthe closed points.3.1. The geometry of P N K and O P N K (1) . Recall that, the projective N -space P N K is obtained by pro-jectivizing the Euclidian ( N + K N + , i.e. is defined as the set of lines passing through theorigin: P N K : = P (cid:0) K N + (cid:1) : = K N + (cid:15) { } (cid:14) ∼ , (19)where the quotient relation ∼ for z ∈ K N + \{ } is: z ∼ λ z ( ∀ λ ∈ K × ) . On P N K , there is the so-called tautological line bundle O P N K ( − z ] ∈ P N K hasfibre: O P N K ( − (cid:12)(cid:12)(cid:12) [ z ] : = K · z ⊂ K N + . ts dual line bundle is the well known: O P N K (1) : = O P N K ( − ∨ . The geometry of P ( Ω P N K ) and O P ( Ω P N K ) (1) . For every point [ z ] ∈ P N K , the tangent space of P N K at[ z ] is: T P N K (cid:12)(cid:12)(cid:12) [ z ] = K N + (cid:14) K · z , and the total tangent space of P N K : T P N K : = T hor K N + (cid:14) ∼ , (20)is the quotient space of the horizontal tangent space of K N + \ { } :T hor K N + : = (cid:110) ( z , [ ξ ]) : z ∈ K N + \ { } and [ ξ ] ∈ K N + (cid:14) K · z (cid:111) , (21)by the quotient relation ∼ : ( z , [ ξ ]) ∼ ( λ z , [ λξ ]) ( ∀ λ ∈ K × ) . λ z ξ C N + z (cid:98) X λξ Now, the K -variety associated to P ( Ω P N K ) is just the projectivized tangent space P (T P N K ), which isobtained by projectivizing each tangent space T P N K (cid:12)(cid:12)(cid:12) [ z ] at every point [ z ] ∈ P N K : P (T P N K ) (cid:12)(cid:12)(cid:12) [ z ] : = P (cid:0) T P N K (cid:12)(cid:12)(cid:12) [ z ] (cid:1) . And the Serre line bundle O P ( Ω P N K ) (1) on P ( Ω P N K ) corresponds to the ‘Serre line bundle’ O P (T P N K ) (1)on P (T P N K ), which after restricting on P (T P N K ) (cid:12)(cid:12)(cid:12) [ z ] becomes O P (T P N K | [ z ] ) (1). In other words, the Serre linebundle O P (T P N K ) (1) is the dual of the tautological line bundle O P (T P N K ) ( − z ] , [ ξ ]) ∈ P (T P N K ), has fibre: O P (T P N K ) ( − (cid:12)(cid:12)(cid:12) ([ z ] , [ ξ ]) : = K · [ ξ ] ⊂ T P N K (cid:12)(cid:12)(cid:12) [ z ] = K N + (cid:14) K · z . .3. The geometry of P ( Ω V / P ♦ K ) , P and P t . Recalling (6), the K -variety V associated to V ⊂ P ♦ K × K P N K is: V : = (cid:110)(cid:0) [ F , . . . , F c + r ] , [ z ] (cid:1) ∈ P ♦ K × P N K : F i ( z ) = , ∀ i = · · · c (cid:111) . Moreover, recalling (11), the K -variety: P (T V / P ♦ K ) ⊂ P ♦ K × P (T P N K )associated to P ( Ω V / P ♦ K ) ⊂ P ♦ K × K P ( Ω P N K ) is: P (T V / P ♦ K ) : = (cid:110)(cid:0) [ F , . . . , F c + r ] , ([ z ] , [ ξ ]) (cid:1) : F i ( z ) = , dF i (cid:12)(cid:12)(cid:12) z ( ξ ) = , ∀ i = · · · c (cid:111) . Similarly, the K -variety: P ⊂ P (T V / P ♦ K )associated to P ⊂ P ( Ω V / P ♦ K ) is: P : = (cid:110)(cid:0) [ F , . . . , F c + r ] , ([ z ] , [ ξ ]) (cid:1) : F i ( z ) = , dF j (cid:12)(cid:12)(cid:12) z ( ξ ) = , ∀ i = · · · c + r , ∀ j = · · · c (cid:111) , and the K -variety: F c + ,..., F c + r P F ,..., F c ⊂ P (T P N K )associated to (18) is: F c + ,..., F c + r P F ,..., F c : = (cid:110) ([ z ] , [ ξ ]) : F i ( z ) = , dF j (cid:12)(cid:12)(cid:12) z ( ξ ) = , ∀ i = · · · c + r , ∀ j = · · · c (cid:111) . (22)Now, the K -variety P t of P t is: P t : = { t } × F c + ,..., F c + r P F ,..., F c . Some hints on the Ampleness Theorem 1.3
The first three Subsections 4.1–4.3 consist of some standard knowledge in algebraic geometry,and the last Subsection 4.4 presents a helpful nefness criterion which suits our moving coe ffi cientsmethod.4.1. Ampleness is Zariski open.
The foundation of our approach is the following classical theo-rem of Grothendieck (cf. [33, III.4.7.1] or [40, p. 29, Theorem 1.2.17]).
Theorem 4.1. [Amplitude in families]
Let f : X → T be a proper morphism of schemes, and let L be a line bundle on X. For every point t ∈ T , denote by:X t : = f − ( t ) , L t : = L (cid:12)(cid:12)(cid:12) X t . Assume that, for some point ∈ T , L is ample on X . Then in T , there is a Zariski open set Ucontaining such that L t is ample on X t , for all t ∈ U. Note that in (13), π = pr ◦ π is a composition of two proper morphisms, hence is proper, and sois (16). Therefore, by virtue of the above theorem, we only need to find one (closed) point t ∈ P ♦ K such that: O P t (1) is ample on P t . (23) .2. Largely twisted Serre line bundle is ample.
Let: π : P ( Ω P N K ) → P N K be the canonical projection. [37, p. 161, Proposition 7.10] yields that, for all su ffi ciently largeinteger (cid:96) , the twisted line bundle below is ample on P ( Ω P N K ): O P ( Ω P N K ) (1) ⊗ π ∗ O P N K ( (cid:96) ) . (24)Recalling (10) and (13), and noting that: π = π ◦ (cid:101) π , (25)for the following ample line bundle H on the scheme P ♦ K × K P N K : H : = pr ∗ O P ♦ K (1) ⊗ pr ∗ O P N K (1) , the twisted line bundle below is ample on P ♦ K × K P ( Ω P N K ): (cid:101) π ∗ O P ( Ω P N K ) (1) ⊗ (cid:101) π ∗ H (cid:96) = (cid:101) π ∗ O P ( Ω P N K ) (1) ⊗ (cid:101) π ∗ (cid:16) pr ∗ O P ♦ K ( (cid:96) ) ⊗ pr ∗ O P N K ( (cid:96) ) (cid:17) = (cid:101) π ∗ O P ( Ω P N K ) (1) ⊗ (pr ◦ (cid:101) π ) ∗ O P ♦ K ( (cid:96) ) ⊗ (pr ◦ (cid:101) π ) ∗ O P N K ( (cid:96) ) [use (13) ] = (cid:101) π ∗ O P ( Ω P N K ) (1) ⊗ π ∗ O P ♦ K ( (cid:96) ) ⊗ π ∗ O P N K ( (cid:96) ) (26) [use (25) ] = (cid:101) π ∗ O P ( Ω P N K ) (1) ⊗ π ∗ O P ♦ K ( (cid:96) ) ⊗ (cid:101) π ∗ (cid:0) π ∗ O P N K ( (cid:96) ) (cid:1) = π ∗ O P ♦ K ( (cid:96) ) (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) ample ⊗ (cid:101) π ∗ (cid:0) O P ( Ω P N K ) (1) ⊗ π ∗ O P N K ( (cid:96) ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ample on P ( Ω P N K ) (cid:1) . In particular, for every point t ∈ P ♦ K , recalling (15), (17), restricting (26) to the subscheme: P t = π − ( t ) ∩ P , we receive an ample line bundle: O P t (1) ⊗ π ∗ O P ♦ K ( (cid:96) ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) trivial line bundle ⊗ π ∗ O P N K ( (cid:96) ) = O P t (1) ⊗ π ∗ O P N K ( (cid:96) ) . (27)4.3. Nefness of negatively twisted cotangent sheaf su ffi ces. As we mentioned at the end ofSubsection 4.1, our goal is to show the existence of one such (closed) point t ∈ P ♦ K satisfying (23).In fact, we can relax this requirement thanks to the following theorem. Theorem 4.2.
For every point t ∈ P ♦ K , the following properties are equivalent. (i) O P t (1) is ample on P t . (ii) There exist two positive integers a , b (cid:62) such that O P t ( a ) ⊗ π ∗ O P N K ( − b ) is ample on P t . (iii) There exist two positive integers a , b (cid:62) such that O P t ( a ) ⊗ π ∗ O P N K ( − b ) is nef on P t . In fact, (cid:96) (cid:62) roof. It is clear that (i) = ⇒ (ii) = ⇒ (iii) , and we now show that (iii) = ⇒ (i) .In fact, the nefness of the negatively twisted Serre line bundle: S at ( − b ) : = O P t ( a ) ⊗ π ∗ O P N K ( − b ) (28)implies that: (27) ⊗ b (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) ample ⊗ (28) ⊗ (cid:96) (cid:124)(cid:123)(cid:122)(cid:125) nef = O P t ( b + a (cid:96) ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ample ! = O P t (1) (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) ample ⊗ ( b + a (cid:96) ) is also ample, because of the well known fact that “ample ⊗ nef = ample” (cf. [40, p. 53, Corollary1.4.10]). (cid:3) By definition, the nefness of (28) means that for every irreducible curve C ⊂ P t , the intersectionnumber C · S at ( − b ) is (cid:62)
0. Recalling now the classical result [37, p. 295, Lemma 1.2], we onlyneed to show that the line bundle S at ( − b ) has a nonzero section on the curve C : H (cid:0) C , S at ( − b ) (cid:1) (cid:44) { } . (29)To this end, of course we like to construct su ffi ciently many global sections: s , . . . , s m ∈ H (cid:0) P t , S at ( − b ) (cid:1) such that their base locus is empty or discrete, whence one of s (cid:12)(cid:12)(cid:12) C , . . . , s m (cid:12)(cid:12)(cid:12) C su ffi ces to con-clude (29).More flexibly, we have: Theorem 4.3.
Suppose that there exist m (cid:62) nonzero sections of certain negatively twisted Serreline bundles: s i ∈ H (cid:0) P t , S a i t ( − b i ) (cid:1) ( i = ··· m ; a i , b i (cid:62) such that their base locus is discrete or empty: dim ∩ mi = BS ( s i ) (cid:54) , then for all positive integers a , b with: ab (cid:62) max (cid:8) a b , . . . , a m b m (cid:9) , the twisted Serre line bundle S at ( − b ) is nef.Proof. For every irreducible curve C ⊂ P t , noting that: C (cid:124)(cid:123)(cid:122)(cid:125) dim = (cid:49) ∩ mi = BS ( s i ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) dim (cid:54) , there exists some integer 1 (cid:54) i (cid:54) m such that: C (cid:49) BS ( s i ) . Therefore s i (cid:12)(cid:12)(cid:12) C is a nonzero section of S a i t ( − b i ) on the curve C : s i ∈ H (cid:0) C , S a i t ( − b i ) (cid:1) \ { } , and hence: C · S a i t ( − b i ) (cid:62) . hus we have the estimate:0 (cid:54) C · (cid:0) S a i t ( − b i ) (cid:1) ⊗ a [ = a C · S a i t ( − b i ) ] = C · O P t ( a i a ) ⊗ π ∗ O P N K ( − b i a ) [see (28) ] = a i C · (cid:0) O P t ( a ) ⊗ π ∗ O P N K ( − b ) (cid:1) − ( b i a − a i b ) C · π ∗ O P N K (1) = a i C · (cid:0) S at ( − b ) (cid:1) − b b i ( a / b − a i / b i (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:62) ) C · π ∗ O P N K (1) . Noting that O P N K (1) is nef and hence is π ∗ O P N K (1) (cf. [40, p. 51, Example 1.4.4]), the above estimateimmediately yields: C · (cid:0) S at ( − b ) (cid:1) (cid:62) b b i a i ( a / b − a i / b i ) C · π ∗ O P N K (1) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:62) (cid:62) . (cid:3) Repeating the same reasoning as in the above two theorems, we obtain:
Proposition 4.4.
For every point t ∈ P ♦ K , if O P t ( (cid:96) ) ⊗ π ∗ O P N K ( − (cid:96) ) is nef on P t for some positiveintegers (cid:96) , (cid:96) (cid:62) , then for any positive integers (cid:96) (cid:48) , (cid:96) (cid:48) (cid:62) with (cid:96) (cid:48) /(cid:96) (cid:48) < (cid:96) /(cid:96) , the twisted linebundle O P t ( (cid:96) (cid:48) ) ⊗ π ∗ O P N K ( − (cid:96) (cid:48) ) is ample on P t . (cid:3) A practical nefness criterion.
However, in practice, it is often di ffi cult to gather enoughglobal sections (with discrete base locus) to guarantee nefness of a line bundle. We need to bemore clever to improve such a coarse nefness criterion with the help of nonzero sections of thesame bundle restricted to proper subvarieties. First, let us introduce the theoretical reason behind. Definition 4.5.
Let X be a variety, and let Y ⊂ X be a subvariety. A line bundle L on X is saidto be nef outside Y if, for every irreducible curve C ⊂ X with C (cid:49) Y , the intersection number C · L (cid:62) L is nef on X if and only if L is nef outside the empty set ∅ ⊂ X . Theorem 4.6 ( Nefness Criterion).
Let X be a noetherian variety, and let L be a line bundle onX. Assume that there exists a set V of closed subvarieties of X satisfying: (i) ∅ ∈ V and X ∈ V ; (ii) for every element Y ∈ V with Y (cid:44) ∅ , there exist finitely many elements Z , . . . , Z (cid:91) ∈ V withZ , . . . , Z (cid:91) (cid:36) Y such that the restricted line bundle L (cid:12)(cid:12)(cid:12) Y is nef outside the union Z ∪ · · · ∪ Z (cid:91) .Then L is nef on X.Proof. For every irreducible curve C ⊂ X , we have to show that C · L (cid:62) C · L <
0. Then introduce the subset N ⊂ V consisting of allsubvarieties Y ∈ V which contain the curve C . Clearly, N (cid:51) X , so N is nonempty. Note thatthere is a natural partial order ‘ < ’ on N given by the strict inclusion relation ‘ (cid:36) ’. Since X isnoetherian, N has a minimum element M ⊃ C . We now show a contradiction.In fact, according to (ii) , there exist some elements V (cid:51) Z , . . . , Z (cid:91) (cid:36) M such that L (cid:12)(cid:12)(cid:12) M is nefoutside Z ∪ · · · ∪ Z (cid:91) . Rembering that:0 > C · L = C · L (cid:12)(cid:12)(cid:12) M , he curve C is forced to lie in the union Z ∪ · · · ∪ Z (cid:91) , and thanks to irreducibility, it is furthermorecontained in one certain: Z i (cid:124)(cid:123)(cid:122)(cid:125) (cid:36) M ∈ N , which contradicts the minimality of M ! (cid:3) Now, using the same idea as around (29), we may realize (ii) above with the help of sectionsover proper subvarieties.
Corollary 4.7.
Let X be a noetherian variety, and let L be a line bundle on X. Assume that thereexists a set V of closed subvarieties of X satisfying: (i) ∅ ∈ V and X ∈ V ; (ii’) every element ∅ (cid:44) Y ∈ V is a union of some elements Y , . . . , Y (cid:11) ∈ V such that the unionof base loci: ∪ (cid:11) • = (cid:16) ∩ s ∈ H ( Y • , L | Y • ) (cid:8) s = (cid:9)(cid:17) is contained in a union of some elements V (cid:51) Z , . . . , Z (cid:91) (cid:36) Y, except discrete points.Then L is nef on X. (cid:3) A proof blueprint of the Ampleness Theorem 1.3
Main Nefness Theorem.
Recalling Theorem 4.2, the Ampleness Theorem 1.3 is a conse-quence of the theorem below, whose e ff ective bound d ( (cid:114) ) for (cid:114) = Theorem 5.1.
Given any positive integer (cid:114) (cid:62) , there exists a lower degree bound d ( (cid:114) ) (cid:29) suchthat, for all degrees d , . . . , d c + r (cid:62) d ( (cid:114) ) , for a very generic t ∈ P ♦ K , the negatively twisted Serreline bundle O P t (1) ⊗ π ∗ O P N K ( − (cid:114) ) is nef on P t . It su ffi ces to find one such t ∈ P ♦ K to guarantee‘very generic’ (cf. [40, p. 56, Proposition 1.4.14]).We will prove Theorem 5.1 in two steps. At first, in Subsection 5.2, we sketch the proof in thecentral cases when all c + r hypersurfaces are approximately of the same large degrees. Then, inSubsection 5.3, we play a product coup to embrace all large degrees.5.2. The central cases of relatively the same large degrees.Theorem 5.2.
For any fixed c + r positive integers (cid:15) , . . . , (cid:15) c + r (cid:62) , for every su ffi ciently largeinteger d (cid:29) , Theorem 5.1 holds with d i = d + (cid:15) i , i = · · · c + r. When c + r (cid:62) N , generically X is discrete or empty, so there is nothing to prove. Assuming c + r (cid:54) N −
1, we now outline the proof.
Step 1.
In the entire family of c + r hypersurfaces with degrees d + λ , . . . , d + λ c + r , whoseprojective parameter space is P ♦ K (see (3)), we select a specific subfamily which best suits ourmoving coe ffi cients method, whose projective parameter space is a subvariety: P (cid:169) K ⊂ P ♦ K [see (140) ] . For the details of this subfamily, see Subsection 7.1. t ∈ P ♦ K \ ∪ ∞ i = Z i for some countable proper subvarieties Z i (cid:36) P ♦ K . ecalling (5) and (7), we then consider the subfamily of intersections Y ⊂ X :pr − (cid:16) P (cid:169) K (cid:17) ∩ X = : Y ⊂ P (cid:169) K × K P N K = pr − (cid:16) P (cid:169) K (cid:17) . Recalling (13), (14), we introduce the subscheme of P : P (cid:48) : = (cid:101) π − ( Y ) ∩ P ⊂ P ♦ K × K P ( Ω P N K ) , which is parametrized by Y . By restriction, (13) yields the commutative diagram: P (cid:48) π = pr ◦ (cid:101) π (cid:6) (cid:6) π = pr ◦ (cid:101) π (cid:24) (cid:24) (cid:101) π (cid:15) (cid:15) Y pr (cid:126) (cid:126) pr (cid:32) (cid:32) P (cid:169) K P N K . (30)Introducing the restricted Serre line bundle O P (cid:48) (1) : = O P (1) (cid:12)(cid:12)(cid:12) P (cid:48) over P (cid:48) , in order to establishTheorem 5.2, it su ffi ces to provide one such example. In fact, we will prove Theorem 5.3.
For a generic closed point t ∈ P (cid:169) K , the bundle O P (cid:48) t (1) ⊗ π ∗ O P N K ( − (cid:114) ) is nef on P (cid:48) t : = P t . Step 2.
The central objects now are the universal negatively twisted Serre line bundles : O P (cid:48) ( a , b , − c ) : = O P (cid:48) ( a ) ⊗ π ∗ O P (cid:169) K ( b ) ⊗ π ∗ O P N K ( − c ) , where a , c are positive integers such that c / a (cid:62) (cid:114) , and where b are any integers.Taking advantage of the moving coe ffi cients method, firstly, we construct a series of globaluniversal negatively twisted symmetric di ff erential n -forms: S (cid:96) ∈ Γ (cid:0) P (cid:48) , O P (cid:48) ( n , N , − ♥ (cid:96) ) (cid:1) ( (cid:96) = ··· (cid:100) ) , (31)where n : = N − ( c + r ) (cid:62) ♥ (cid:96) / n (cid:62) (cid:114) , and where we always use the symbol ‘ (cid:100) ’ to denoteauxiliary positive integers, which vary according to the context.Secondly, for every integer 1 (cid:54) η (cid:54) n −
1, for every sequence of ascending indices :0 (cid:54) v < · · · < v η (cid:54) N , considering the vanishing part of the corresponding η coordinates: v ,..., v η P (cid:48) : = P (cid:48) ∩ π − { z v = · · · = z v η = } (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : v ,..., v η P N , we construct a series of universal negatively twisted symmetric di ff erential ( n − η )-forms on it: v ,..., v η S (cid:96) ∈ Γ (cid:0) v ,..., v η P (cid:48) , O P (cid:48) ( n − η, N − η, − v ,..., v η ♥ (cid:96) ) (cid:1) ( (cid:96) = ··· (cid:100) ) . (32)where all v ,..., v η ♥ (cid:96) / ( n − η ) (cid:62) (cid:114) .This step will be accomplished in Sections 6 and 7. Step 3.
From now on, we view every scheme as its K -variety.Firstly, we control the base locus of all the global sections obtained in (31): BS : = Base Locus of { S (cid:96) } (cid:54) (cid:96) (cid:54) (cid:100) ⊂ P (cid:48) . n fact, on the coordinates nonvanishing part of P (cid:48) : P (cid:48)◦ : = P (cid:48) ∩ π − { z · · · z N (cid:44) } , we prove that: dim BS ∩ P (cid:48)◦ (cid:54) dim P (cid:169) K . (33)Secondly, we control the base locus of all the sections obtained in (32): v ,..., v η BS : = Base Locus of { v ,..., v η S (cid:96) } (cid:54) (cid:96) (cid:54) (cid:100) ⊂ v ,..., v η P (cid:48) . In fact, on the corresponding ‘coordinates nonvanishing part’ of v ,..., v η P (cid:48) : v ,..., v η P (cid:48)◦ : = v ,..., v η P (cid:48) ∩ π − { z r · · · z r N − η (cid:44) } , where: { r , . . . , r N − η } : = { , . . . , N } \ { v , . . . , v η } , (34)we prove that: dim v ,..., v η BS ∩ v ,..., v η P (cid:48)◦ (cid:54) dim P (cid:169) K . (35)This crucial step will be accomplished in Sections 9 and 10. Anticipating, we would like to em-phasize that, in order to lower down dimensions of base loci for global symmetric di ff erential forms(or for higher order jet di ff erential forms in Kobayashi hyperbolicity conjecture), a substantialamount of algebraic geometry work is required, mainly because some already known (cid:14) constructedsections have the annoying tendency to proliferate by multiplying each other without shrinkingtheir base loci (0 × anything = ffi culty is to devise a wealth of inde-pendent symmetric di ff erential forms, which the Moving Coe ffi cients Method is designed for , andthe second main di ffi culty is to establish the emptiness (cid:14) discreteness of their base loci, an ultimatedi ffi culty that will be settled in the Core Lemma
Step 4.
Firstly, for the regular map: π : P (cid:48) −→ P (cid:169) K , noting the dimension estimates (33), (35) of the base loci, applying now a classical theorem [35,p. 132, Theorem 11.12], we know that there exists a proper closed algebraic subvariety: Σ (cid:36) P (cid:169) K such that, for every closed point t outside Σ : t ∈ P (cid:169) K \ Σ , (i) the base locus of the restricted symmetric di ff erential n -forms: BS t : = Base Locus of (cid:8) S (cid:96) ( t ) : = S (cid:96) (cid:12)(cid:12)(cid:12) P (cid:48) t (cid:9) (cid:54) (cid:96) (cid:54) (cid:100) ⊂ P (cid:48) t is discrete or empty over the coordinates nonvanishing part: dim BS t ∩ P (cid:48)◦ t (cid:54) , (36)where: P (cid:48)◦ t : = P (cid:48)◦ ∩ π − ( t ); ii) the base locus of the restricted symmetric di ff erential ( n − η )-forms: v ,..., v η BS t : = Base Locus of (cid:110) v ,..., v η S (cid:96) ( t ) : = v ,..., v η S (cid:96) (cid:12)(cid:12)(cid:12) v ,..., v η P (cid:48) t (cid:111) (cid:54) (cid:96) (cid:54) (cid:100) ⊂ v ,..., v η P (cid:48) t is discrete or empty over the corresponding ‘coordinates nonvanishing part’: dim v ,..., v η BS t ∩ v ,..., v η P (cid:48)◦ t (cid:54) , (37)where: v ,..., v η P (cid:48)◦ t : = v ,..., v η P (cid:48)◦ ∩ π − ( t ) . Secondly, there exists a proper closed algebraic subvariety: Σ (cid:48) (cid:36) P (cid:169) K such that, for every closed point t outside Σ (cid:48) : t ∈ P (cid:169) K \ Σ (cid:48) , the fibre: Y t : = Y ∩ pr − ( t )is smooth and of dimension n = N − ( c + r ), and it satisfies: dim Y t ∩ pr − (cid:0) v ,..., v n P N (cid:1) = (0 (cid:54) v < ··· < v n (cid:54) N ) , (38)i.e. the intersection of Y t — (under the regular map pr ) viewed as a dimension n subvariety in P N — with every n coordinate hyperplanes: v ,..., v n P N : = { z v = · · · = z v n = } is just finitely many points, which we denote by: v ,..., v n Y t (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) < ∞ ⊂ v ,..., v n P N . (39)Now, we shall conclude Theorem 5.3 for every closed point t ∈ P (cid:169) K \ ( Σ ∪ Σ (cid:48) ). Proof of Theorem 5.3.
For the line bundle L = O P (cid:48) t (1) ⊗ π ∗ O P N K ( − (cid:114) ) over the variety P (cid:48) t , we claimthat the set of subvarieties: V : = (cid:110) ∅ , P (cid:48) t , v ,..., v η P (cid:48) t (cid:111) (cid:54) η (cid:54) n (cid:54) v < ··· < v η (cid:54) N satisfies the conditions of Theorem 4.6.Indeed, firstly, recalling (36), the sections { S (cid:96) ( t ) } (cid:96) = ··· (cid:100) have empty (cid:14) discrete base locus over thecoordinates nonvanishing part, i.e. outside ∪ Nj = j P (cid:48) t . Hence, using an adaptation of Theorem 4.3,remembering (cid:114) / (cid:54) min {♥ (cid:96) / n } (cid:54) (cid:96) (cid:54) (cid:100) , the line bundle O P (cid:48) t (1) ⊗ π ∗ O P N ( − (cid:114) ) is nef outside ∪ Nj = j P (cid:48) t .Secondly, for every integer η = · · · n −
1, recalling the dimension estimate (37), again byTheorem 4.3, remembering (cid:114) / (cid:54) min { v ,..., v η ♥ (cid:96) / ( n − η ) } (cid:54) (cid:96) (cid:54) (cid:100) , the line bundle O P (cid:48) t (1) ⊗ π ∗ O P N ( − (cid:114) )is nef on v ,..., v η P (cid:48) t outside ∪ N − η j = v ,..., v η , r j P (cid:48) t (see (34)).Lastly, for η = n , noting that under the projection π : P (cid:48) t → Y t , thanks to (38), every subvari-ety v ,..., v n P (cid:48) t contracts to discrete points v ,..., v n Y t , we see that on v ,..., v n P (cid:48) t , the line bundle O P (cid:48) t (1) ⊗ π ∗ O P N K ( − (cid:114) ) (cid:27) O P (cid:48) t (1) is not only nef, but also ample!Summarizing the above three parts, by Theorem 4.6, we conclude the proof. (cid:3) .3. Product Coup.
We will use in an essential way Theorem 5.2 with all (cid:15) i equal to either 1 or2. To begin with, we need an elementary Observation 5.4.
For all positive integers d (cid:62) , every integer d (cid:62) d + d is a sum of nonnegativemultiples of d + and d + .Proof. According to the Euclidian division, we can write d as: d = p ( d + + q for some positive integer p (cid:62) (cid:54) q (cid:54) d . We claim that p (cid:62) q .Otherwise, we would have: p (cid:54) q − (cid:54) d − , which would imply the estimate: d = p ( d + + q (cid:54) ( d −
1) ( d + + d = d + d − , contradicting our assumption.Therefore, we can write d as: d = ( p − q ) (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) (cid:62) ( d + + q ( d + , which concludes the proof. (cid:3) Proof of Theorem 5.1.
Take one su ffi ciently large integer d such that Theorem 5.2 holds for anyintegers (cid:15) i ∈ { , } , i = · · · c + r . Now, the above observation says that all large degrees d , . . . , d c + r (cid:62) d + d can be written as d i = p i ( d + + q i ( d + p i , q i (cid:62) i = · · · c + r . Let F i : = f i · · · f ip i f ip i + · · · f ip i + q i be a product of some p i homo-geneous polynomials f i , . . . , f ip i each of degree d + q i homogeneous polynomials f ip i + , . . . , f ip i + q i each of degree d +
2, so that F i has degree d i .Recalling (22), a point ([ z ] , [ ξ ]) ∈ P (T P N K ) lies in F c + ,..., F c + r P F ,..., F c if and only if: F i ( z ) = , dF j (cid:12)(cid:12)(cid:12) z ( ξ ) = ( ∀ i = ··· c + r , ∀ j = ··· c ) . Note that, for every j = · · · c , the pair of equations: F j ( z ) = , dF j (cid:12)(cid:12)(cid:12) z ( ξ ) = ∃ (cid:54) v j (cid:54) p j + q j s . t . f jv j ( z ) = , d f jv j (cid:12)(cid:12)(cid:12) z ( ξ ) = , (41)or to: ∃ (cid:54) w j < w j (cid:54) p j + q j s . t . f jw j ( z ) = , f jw j ( z ) = . (42)Therefore, ([ z ] , [ ξ ]) ∈ F c + ,..., F c + r P F ,..., F c is equivalent to say that there exists a subset { i , . . . , i k } ⊂{ , . . . , c } of cardinality k ( k = ∅ ) such that, firstly, for every index j ∈ { i · · · i k } , ( z , ξ ) is asolution of (40) of type (41), secondly, for every index j ∈ { , . . . , c } \ { i · · · i k } , ( z , ξ ) is a solutionof (40) of type (42), and lastly, for every j = c + · · · c + r , one of f j , . . . , f jp j + q j vanishes at z . Thus,we see that the variety F c + ,..., F c + r P F ,..., F c actually decomposes into a union of subvarieties: F c + ,..., F c + r P F ,..., F c = ∪ k = ··· c ∪ (cid:54) i < ··· < i k (cid:54) c ∪ (cid:54) v ij (cid:54) p vj + q vj j = ··· k ∪ { r ,..., r c − k } = { ,..., c }\{ i ,..., i k } (cid:54) w rl < w rl (cid:54) p rl + q rl l = ··· c − k ∪ (cid:54) u j (cid:54) p j + q j j = c + ··· c + rf r w r , f r w r ,..., f rc − kw rc − k , f rc − kw rc − k , f c + uc + ,..., f c + ruc + r P f i vi ,..., f ikvik . imilarly, we can show that the scheme F c + ,..., F c + r P F ,..., F c also decomposes into a union of sub-schemes: F c + ,..., F c + r P F ,..., F c = ∪ k = ··· c ∪ (cid:54) i < ··· < i k (cid:54) c ∪ (cid:54) v ij (cid:54) p vj + q vj j = ··· k ∪ { r ,..., r c − k } = { ,..., c }\{ i ,..., i k } (cid:54) w rl < w rl (cid:54) p rl + q rl l = ··· c − k ∪ (cid:54) u j (cid:54) p j + q j j = c + ··· c + rf r w r , f r w r ,..., f rc − kw rc − k , f rc − kw rc − k , f c + uc + ,..., f c + ruc + r P f i vi ,..., f ikvik . (43)Note that, for each subscheme on the right hand side, the number of polynomials on the lower-leftof ‘ P ’ is L = c − k ) + r , and the number of polynomials on the lower-right is R = k , whence 2 R + L = c + r (cid:62) N . Now, applying Theorem 5.2, we can choose one { f •• } • , • such that the twisted Serreline bundle O P ( Ω P N K ) (1) ⊗ π ∗ O P N K ( − (cid:114) ) is nef on each subscheme f r w r , f r w r ,..., f rc − kw rc − k , f rc − kw rc − k , f c + uc + ,..., f c + ruc + r P f i vi ,..., f ikvik ,and therefore is also nef on their union F c + ,..., F c + r P F ,..., F c . Since nefness is a very generic propertyin family, we conclude the proof. (cid:3) Generalization of Brotbek’s symmetric di ff erentials forms Preliminaries on symmetric di ff erential forms in projective space. For a fixed algebraicallyclosed field K , for three fixed integers N , c , r (cid:62) N (cid:62)
2, 2 c + r (cid:62) N and c + r (cid:54) N − c + r positive integers d , . . . , d c + r , let: H i ⊂ P N K ( i = ··· c + r ) be c + r hypersurfaces defined by some degree d i homogeneous polynomials: F i ∈ K [ z , . . . , z N ] , let V be the intersection of the first c hypersurfaces: V : = H ∩ · · · ∩ H c = (cid:8) [ z ] ∈ P N K : F i ( z ) = , ∀ i = · · · c (cid:9) , (44)and let X be the intersection of all the c + r hypersurfaces: X : = H ∩ · · · ∩ H c (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = V ∩ H c + ∩ · · · ∩ H c + r (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) r more hypersurfaces = (cid:8) [ z ] ∈ P N K : F i ( z ) = , ∀ i = · · · c + r (cid:9) . (45)It is well known that, for generic choices of { F i } c + ri = , the intersection V = ∩ ci = H i and X = ∩ c + ri = H i are both smooth complete, and we shall assume this henceforth. In Subsections 6.1–6.4, we fo-cus on smooth K -varieties to provide a geometric approach to generalize Brotbek’s symmetricdi ff erential forms, where the ambient field K is assumed to be algebraically closed. In addition,in Subsection 6.5, we will give another quick algebraic approach, without any assumption on theambient field K .Recalling (19), let us denote by: π : K N + \ { } −→ P N K the canonical projection.For every integer k , the standard twisted regular function sheaf O P N K ( k ), geometrically, can bedefined as, for all Zariski open subset U in P N K , the corresponding section set Γ (cid:0) U , O P N K ( k ) (cid:1) consists f all the regular functions (cid:98) f on π − ( U ) satisfying: (cid:98) f ( λ z ) = λ k (cid:98) f ( z ) ( ∀ z ∈ π − ( U ) , λ ∈ K × ) . (46)For the cone (cid:98) V : = π − ( V ) of V : (cid:98) V = (cid:8) z ∈ K N + \ { } : F i ( z ) = , ∀ i = · · · c (cid:9) , recalling (20), (21), we can similarly define its horizontal tangent bundle T hor (cid:98) V which has fibre atany point z ∈ (cid:98) V : T hor (cid:98) V (cid:12)(cid:12)(cid:12) z = (cid:8) [ ξ ] ∈ K N + (cid:46) K · z : dF i (cid:12)(cid:12)(cid:12) z ( ξ ) = , ∀ i = · · · c (cid:9) . Its total space is:T hor (cid:98) V : = (cid:8) ( z , [ ξ ]) : z ∈ (cid:98) V , [ ξ ] ∈ K N + (cid:46) K · z , dF i (cid:12)(cid:12)(cid:12) z ( ξ ) = , ∀ i = · · · c (cid:9) . (47)Then similarly we receive the total tangent bundle T V of V as:T V = T hor (cid:98) V / ∼ , where ( z , [ ξ ]) ∼ ( λ z , [ λξ ]) , ∀ λ ∈ K × . Let Ω V be the dual bundle of T V , i.e. the cotangent bundle of V , and let Ω hor (cid:98) V be the dualbundle of T hor (cid:98) V . For all positive integers l (cid:62) ♥ ∈ Z , we use the standard notation Sym l Ω V to denote the symmetric l -tensor-power of the vector bundle Ω V , and we use Sym l Ω V ( ♥ )to denote the twisted vector bundle Sym l Ω V ⊗ O V ( ♥ ). Proposition 6.1.
For two fixed integers l (cid:62) , ♥ ∈ Z , and for every Zariski open set U ⊂ V togetherwith its cone (cid:98) U : = π − ( U ) , there is a canonical injection: Γ (cid:0) U , Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) , whose image is the set of sections Φ enjoying: Φ (cid:0) λ z , [ λξ ] (cid:1) = λ ♥ Φ (cid:0) z , [ ξ ] (cid:1) , (48) for all z ∈ (cid:98) U, for all [ ξ ] ∈ T hor (cid:98) V (cid:12)(cid:12)(cid:12) z and for all λ ∈ K × .Proof. Note that we have two canonical injections of vector bundles: π ∗ Sym l Ω V (cid:44) → Sym l Ω hor (cid:98) V ,π ∗ O V ( ♥ ) (cid:44) → O (cid:98) V , since the tensor functor is left exact (torsion free) in the category of K -vector bundles, the tensoringof the above two injections remains an injection: π ∗ Sym l Ω V ⊗ π ∗ O V ( ♥ ) (cid:44) → Sym l Ω hor (cid:98) V ⊗ O (cid:98) V . Recalling that:
Sym l Ω V ( ♥ ) = Sym l Ω V ⊗ O V ( ♥ ) , we can rewrite the above injection as: π ∗ Sym l Ω V ( ♥ ) (cid:44) → Sym l Ω hor (cid:98) V . With U ⊂ X Zariski open, applying the global section functor Γ ( (cid:98) U , · ), which is left exact, wereceive: Γ (cid:0) (cid:98) U , π ∗ Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) . astly, we have an injection: Γ (cid:0) U , Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , π ∗ Sym l Ω V ( ♥ ) (cid:1) , whence, by composing the previous two injections, we conclude: Γ (cid:0) U , Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) . To view explicitly the image of this injection, notice that in the case l =
0, it is the standardinjection: Γ (cid:0) U , O V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , O (cid:98) V ) f (cid:55)→ π ∗ f , whose image consists of, as a consequence of the definition (46) above, all functions (cid:98) f on (cid:98) U satisfying (cid:98) f ( λ z ) = λ ♥ (cid:98) f ( z ), for all z ∈ (cid:98) U and for all λ ∈ K × .Furthermore, in the case ♥ =
0, the image of the injection: Γ (cid:0) U , Sym l Ω V (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) ω (cid:55)→ π ∗ ω, consists of sections (cid:98) ω on (cid:98) U satisfying: (cid:98) ω ( z , [ ξ ]) = (cid:98) ω ( λ z , [ λξ ]) , for all z ∈ (cid:98) U , all [ ξ ] ∈ T hor (cid:98) V (cid:12)(cid:12)(cid:12) z and all λ ∈ K × .As Sym l Ω V ( ♥ ) = Sym l Ω V ⊗ O V ( ♥ ), composing the above two observations by tensoring thecorresponding two injections, we see that any element Φ in the image of the injection: Γ (cid:0) U , Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) , (49)automatically satisfies (48). On the other hand, for every element Φ in Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) satisfying(48), we can construct the corresponding element φ in Γ (cid:0) U , Sym l Ω V ( ♥ ) (cid:1) , which maps to Φ underthe injection (49). (cid:3) Let Y ⊂ V be a regular subvariety. Replacing the underground variety V by Y , in much the sameway we can show: Proposition 6.2.
For two fixed integers l (cid:62) , ♥ ∈ Z , and for every Zariski open set U ⊂ Y togetherwith its cone (cid:98) U : = π − ( U ) , there is a canonical injection: Γ (cid:0) U , Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0) (cid:98) U , Sym l Ω hor (cid:98) V (cid:1) , whose image is the set of sections Φ enjoying: Φ (cid:0) λ z , [ λξ ] (cid:1) = λ ♥ Φ (cid:0) z , [ ξ ] (cid:1) , (50) for all z ∈ (cid:98) U, for all [ ξ ] ∈ T hor (cid:98) V (cid:12)(cid:12)(cid:12) z and for all λ ∈ K × . (cid:3) In future applications, we will mainly interest in the sections: Γ (cid:0) Y , Sym l Ω V ( ♥ ) (cid:1) , where Y = X or Y = X ∩ { z v = } ∩ · · · ∩ { z v η = } for some vanishing coordinate indices0 (cid:54) v < · · · < v η (cid:54) N . .2. Global regular symmetric horizontal di ff erential forms. In our coming applications, wewill be mainly concerned with
Fermat-type hypersursurfaces H i defined by some homogeneouspolynomials F i of the form: F i = N (cid:88) j = A ji z λ j j ( i = ··· c + r ) , (51)where λ , . . . , λ N are some positive integers and where A ji ∈ K [ z , z , . . . , z N ] are some homoge-neous polynomials, with all terms of F i having the same degree:deg A ji + λ j = deg F i = : d i ( i = ··· c + r ; j = ··· N ) . (52)Di ff erentiating F i , we receive: dF i = N (cid:88) j = B ji z λ j − j , (53)where: B ji : = z j dA ji + λ j A ji dz j ( i = ··· c + r ; j = ··· N ) . (54)To make the terms of F i have the same structure as that of dF i , let us denote: (cid:101) A ji : = A ji z j , (55)so that: F i = N (cid:88) j = (cid:101) A ji z λ j − j . Recalling (45), we denote the cone of X by: (cid:98) X = (cid:8) z ∈ K N + \ { } : F i ( z ) = , ∀ i = · · · c + r (cid:9) . For all z ∈ (cid:98) X and [ ξ ] ∈ T hor (cid:98) V (cid:12)(cid:12)(cid:12) z , by the very definition (47) of T hor (cid:98) V , we have: (cid:80) Nj = (cid:101) A ji z λ j − j ( z ) = ( i = ··· c + r ) , (cid:80) Nj = B ji ( z , ξ ) z λ j − j ( z ) = ( i = ··· c ) . (56)For convenience, dropping z , ξ , we rewrite the above equations as: (cid:80) Nj = (cid:101) A ji z λ j − j = ( i = ··· c + r ) , (cid:80) Nj = B ji z λ j − j = ( i = ··· c ) , and formally, we view them as a system of linear equations with respect to the unknown variables z λ − , . . . , z λ N − N , of which the associated coe ffi cient matrix, of size ( c + r + c ) × ( N + C : = (cid:101) A · · · (cid:101) A N ... ... (cid:101) A c + r · · · (cid:101) A Nc + r B · · · B N ... ... B c · · · B Nc , (57) o that the system reads as: C z λ − ... z λ N − N = . (58)Recalling our assumption: n = N − ( c + r ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = dim X (cid:62) , since 2 c + r (cid:62) N , we have 1 (cid:54) n (cid:54) c .Let now D be the upper ( c + r + n (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = N ) × ( N +
1) submatrix of C , i.e. consisting of the first ( c + r + n )rows of C : D : = (cid:101) A · · · (cid:101) A N ... ... (cid:101) A c + r · · · (cid:101) A Nc + r B · · · B N ... ... B n · · · B Nn . (59)For j = · · · N , let (cid:98) D j denote the submatrix of D obtained by omitting the ( j + (cid:98) D j : = (cid:101) A · · · (cid:98)(cid:101) A j . . . (cid:101) A N ... ... (cid:101) A c + r · · · (cid:100)(cid:101) A jc + r . . . (cid:101) A Nc + r B · · · (cid:98) B j . . . B N ... ... B n · · · (cid:98) B jn . . . B Nn , (60)and let D j denote the ( j + D .Denote: W j : = { z j (cid:44) } ⊂ P N ( j = ··· N ) (61)the canonical a ffi ne open subsets, whose cones are: (cid:98) W j : = π − ( W j ) ⊂ K N + \ { } . (62)Denote also: U j : = W j ∩ X (63)the open subsets of X , whose cones are: (cid:98) U j : = π − ( U j ) ⊂ (cid:98) X . (64)Recalling the horizontal tangent bundle of K N + :T hor K N + = (cid:8) ( z , [ ξ ]) : z ∈ K N + \ { } and [ ξ ] ∈ K N + / K · z (cid:9) , now let Ω hor K N + be its dual bundle. roposition 6.3. For every j = · · · N, on the a ffi ne set: (cid:98) W j = { z j (cid:44) } ⊂ K N + \ { } , the following a ffi ne symmetric horizontal di ff erential n-form is well defined: (cid:98) ω j : = ( − j z λ j − j det (cid:0)(cid:98) D j (cid:1) ∈ Γ (cid:16) (cid:98) W j , Sym n Ω hor K N + (cid:17) . (65)The essence of this proposition lies in the famous Euler’s Identity . Lemma 6.4. [Euler’s Identity]
For every homogeneous polynomial P ∈ K [ z , . . . , z N ] , one has: N (cid:88) j = ∂ F ∂ z j · z j = deg F · F , where using di ff erential writes as:dF (cid:12)(cid:12)(cid:12) z ( z ) = : dF ( z , z ) = deg F · F ( z ) , (66) at all points z = ( z , . . . , z N ) ∈ K N + . (cid:3) Proof of Proposition 6.3.
Without loss of generality, we only prove the case j = (cid:101) A ji are regular functions and all B ji are regular1-forms on K N + , we can see without di ffi culty that: (cid:98) ω = z λ − det (cid:0)(cid:98) D (cid:1) = z λ − det (cid:101) A · · · (cid:101) A N ... ... (cid:101) A c + r · · · (cid:101) A Nc + r B · · · B N ... ... B n · · · B Nn ∈ Γ (cid:16)(cid:98) V , Sym n Ω K N + (cid:17) is a well defined regular symmetric di ff erential n -form. Now we need an: Observation 6.5.
Let N (cid:62) be a positive integer, let L be a field with Card L = ∞ , and let F be apolynomial: F ∈ L [ z , . . . , z N ] . Then F is a polynomial without the variable z :F ∈ L [ z , . . . , z N ] ⊂ L [ z , . . . , z N ] if and only if the evaluation map: ev F : L N + −→ L ( x , . . . , x N ) (cid:55)−→ F ( x , . . . , x N ) is independent of the first variable x ∈ L. (cid:3) or the same reason as the above Observation, in order to show that (cid:98) ω descends to a regularsymmetric horizontal di ff erential n -form in Γ (cid:0)(cid:98) V , Sym n Ω hor K N + (cid:1) , we only have to show, at everypoint z ∈ (cid:98) V , for all ξ ∈ T z K N + (cid:27) K N + , λ ∈ K × , that: (cid:98) ω ( z , ξ + λ z ) = (cid:98) ω ( z , ξ ) . (67)In fact, applying Euler’s Identity (66) to the above formula (54), we receive: B ji ( z , z ) = λ j A ji ( z ) dz j ( z , z ) + dA ji ( z , z ) z j ( z ) = λ j A ji ( z ) z j ( z ) + deg A ji · A ji ( z ) z j ( z ) = ( λ j + deg A ji ) (cid:101) A ji ( z ) . Since B ji are 1-forms, we obtain: B ji ( z , ξ + λ z ) = B ji ( z , ξ ) + λ B ji ( z , z ) = B ji ( z , ξ ) + λ ( λ j + deg A ji ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ‘constant’ (cid:101) A ji ( z ) . Therefore, the matrix: (cid:101) A · · · (cid:101) A N ... ... (cid:101) A c + r · · · (cid:101) A Nc + r B · · · B N ... ... B n · · · B Nn ( z , ξ + λ z )not only has the same first c + r rows as the matrix: (cid:101) A · · · (cid:101) A N ... ... (cid:101) A c + r · · · (cid:101) A Nc + r B · · · B N ... ... B n · · · B Nn ( z , ξ ) , but also for (cid:96) = · · · n , the ( c + r + (cid:96) )-th row of the former one equals to the ( c + r + (cid:96) )-th row ofthe latter one plus a multiple of the (cid:96) -th row. Therefore both matrices have the same determinant,which verifies (67). (cid:3) Inspired by the explicit global symmetric di ff erential forms in Lemma 4.5 of Brotbek’s paper [7],we carry out a simple proposition employing the above notation. First, let us recall the well knownCramer’s rule in a less familiar formulation (cf. [39, p. 513, Theorem 4.4]). Theorem 6.6. [Cramer’s rule]
In a commutative ring R, for all positive integers N (cid:62) , let:A , A , . . . , A N ∈ R N be ( N + column vectors, and suppose that z , z , . . . , z N ∈ R satisfy:A z + A z + · · · + A N z N = . (68) hen for all index pairs (cid:54) i < j (cid:54) N, there holds the identity: ( − j det (cid:0) A , . . . , (cid:98) A j , . . . , A N (cid:1) z i = ( − i det (cid:0) A , . . . , (cid:98) A i , . . . , A N (cid:1) z j . (69) Proof.
By permuting the indices, without loss of generality, we may assume i = A z = − N (cid:88) (cid:96) = A (cid:96) z (cid:96) . (70)Hence we may compute the left hand side of (69) as:( − j det (cid:0) A , A , . . . , (cid:98) A j , . . . , A N (cid:1) z = ( − j det (cid:0) A z , A , . . . , (cid:98) A j , . . . , A N (cid:1) [substitute (70) ] = ( − j det (cid:18) − N (cid:88) (cid:96) = A (cid:96) z (cid:96) , A , . . . , (cid:98) A j , . . . , A N (cid:19) = ( − j + N (cid:88) (cid:96) = det (cid:0) A (cid:96) , A , . . . , (cid:98) A j , . . . , A N (cid:1) z (cid:96) [only (cid:96) = j is nonzero] = ( − j + det (cid:0) A j , A , . . . , (cid:98) A j , . . . , A N (cid:1) z j = ( − det (cid:0) (cid:98) A , A , . . . , A N (cid:1) z j , which is exactly the right hand side. (cid:3) Proposition 6.7.
The following ( N + a ffi ne regular symmetric horizontal di ff erential n-forms: (cid:98) ω j : = ( − j z λ j − j det (cid:0)(cid:98) D j (cid:1) ∈ Γ (cid:16) (cid:98) U j , Sym n Ω hor (cid:98) V (cid:17) ( j = ··· N ) glue together to make a regular symmetric horizontal di ff erential n-form on (cid:98) X: ω ∈ Γ (cid:16)(cid:98) X , Sym n Ω hor (cid:98) V (cid:17) . Proof.
Our proof divides into two parts.
Part 1:
To show that these a ffi ne regular symmetric horizontal di ff erential n -forms (cid:98) ω , . . . , (cid:98) ω N are well defined. Part 2:
To show that any two di ff erent a ffi ne regular symmetric horizontal di ff erential n -forms (cid:98) ω j and (cid:98) ω j glue together along the intersection set (cid:98) U j ∩ (cid:98) U j . Proof of Part 1.
The Proposition 6.3 above shows that the: (cid:98) ω j : = ( − j z λ j − j det (cid:0)(cid:98) D j (cid:1) ∈ Γ (cid:16) (cid:98) W j , Sym n Ω hor K N + (cid:17) ( j = ··· N ) , are well defined, where: (cid:98) W j = { z j (cid:44) } ⊂ K N + \ { } . Thanks to the canonical inclusion embedding of vector bundles: (cid:16) (cid:98) U j , T hor (cid:98) V (cid:17) (cid:44) → (cid:16) (cid:98) W j , T hor K N + (cid:17) , a pullback of (cid:98) ω j concludes the first part. roof of Part 2. Recalling the equations (58), in particular, granted that D consists of the first( c + r + n ) rows of C , we have: D z λ − ... z λ N − N = . Now applying the above Cramer’s rule to all the ( N +
1) columns of D and the ( N +
1) values z λ − , . . . , z λ N − N , for every index pair 0 (cid:54) j < j (cid:54) N , we receive:( − j det (cid:0)(cid:98) D j (cid:1) z λ j − j = ( − j det (cid:0)(cid:98) D j (cid:1) z λ j − j . When z j (cid:44) , z j (cid:44)
0, this immediately yields:( − j z λ j − j det (cid:0)(cid:98) D j (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = (cid:98) ω j = ( − j z λ j − j det (cid:0)(cid:98) D j (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = (cid:98) ω j , thus the two a ffi ne symmetric horizontal di ff erential n -forms (cid:98) ω j and (cid:98) ω j glue together along theiroverlap set (cid:98) U j ∩ (cid:98) U j . (cid:3) By permuting the indices, the above Proposition 6.7 can be trivially generalized to, instead ofthe particular upper ( c + r + n ) × ( N +
1) submatrix D , all ( c + r + n ) × ( N +
1) submatrices of C containing the upper c + r rows, as follows.For all n ascending positive integers 1 (cid:54) j < · · · < j n (cid:54) c , denote C j ,..., j n the ( c + r + n ) × ( N + C consisting of the first upper c + r rows and the rows c + r + j , . . . , c + r + j n . Also,for j = · · · N , let (cid:98) C j ,..., j n ; j denote the submatrix of C j ,..., j n obtained by omitting the ( j + Proposition 6.8.
The following ( N + a ffi ne regular symmetric horizontal di ff erential n-forms: (cid:98) ω j ,..., j n ; j : = ( − j z λ j − j det (cid:0)(cid:98) C j ,..., j n ; j (cid:1) ∈ Γ (cid:16) (cid:98) U j , Sym n Ω hor (cid:98) V (cid:17) ( j = ··· N ) glue together to make a global regular symmetric horizontal di ff erential n-form on (cid:98) X. (cid:3) One step further, the above Proposition 6.8 can be generalized to a larger class of ( c + r + n ) × ( N +
1) submatrices of C , as follows.For any two positive integers l (cid:62) l with: l + l = c + r + n , for any two ascending sequences of positive indices:1 (cid:54) i < · · · < i l (cid:54) c + r , (cid:54) j < · · · < j l (cid:54) c satisfying: { j , . . . , j l } ⊂ { i , . . . , i l } , denote C i ,..., i l j ,..., j l the ( c + r + n ) × ( N +
1) submatrix of C consisting of the rows i , . . . , i l and therows c + r + j , . . . , c + r + j l . Also, for j = , . . . , N , let (cid:98) C i ,..., i l j ,..., j l ; j denote the submatrix of C i ,..., i l j ,..., j l obtained by omitting the ( j + y much the same proof of Proposition 6.7, we obtain: Proposition 6.9.
The following N + a ffi ne regular symmetric horizontal di ff erential l -forms: (cid:98) ω i ,..., i l j ,..., j l ; j : = ( − j z λ j − j det (cid:16)(cid:98) C i ,..., i l j ,..., j l ; j (cid:17) ∈ Γ (cid:16) (cid:98) U j , Sym l Ω hor (cid:98) V (cid:17) ( j = ··· N ) glue together to make a global regular symmetric horizontal di ff erential l -form: (cid:98) ω i ,..., i l j ,..., j l ∈ Γ (cid:16)(cid:98) X , Sym l Ω hor (cid:98) V (cid:17) . (cid:3) Global twisted regular symmetric di ff erential forms. Now, using the structure of the aboveexplicit global forms, and applying the previous Proposition 6.2, we receive a result which, in thecase of pure Fermat-type hypersurfaces (1) where all λ = · · · = λ N = (cid:15) are equal, with also equaldeg F = · · · = deg F c = e + (cid:15) , coincides with Brotbek’s Lemma 4.5 in [7]; Brotbek also implicitlyobtained such symmetric di ff erential forms by his cohomological approach. Proposition 6.10.
Under the assumptions and notation of Proposition 6.9, the global regular sym-metric horizontal di ff erential l -form (cid:98) ω i ,..., i l j ,..., j l is the image of a global twisted regular symmetricdi ff erential l -form: ω i ,..., i l j ,..., j l ∈ Γ (cid:0) X , Sym l Ω V ( ♥ ) (cid:1) under the canonical injection as a particular case of Proposition 6.2: Γ (cid:0) X , Sym l Ω V ( ♥ ) (cid:1) (cid:44) → Γ (cid:0)(cid:98) X , Sym l Ω hor (cid:98) V (cid:1) , where the degree: ♥ : = l (cid:88) p = deg F i p + l (cid:88) q = deg F j q − N (cid:88) j = λ j + N + . (71) For all homogeneous polynomials P ∈ Γ (cid:0) P N , O P N (deg P ) (cid:1) , by multiplication, one receives moreglobal twisted regular symmetric di ff erential l -forms:P ω i ,..., i l j ,..., j l ∈ Γ (cid:0) X , Sym l Ω V (deg P + ♥ ) (cid:1) . (cid:3) It is worth to mention that, again by applying Cramer’s rule in linear algebra, one can constructdeterminantal shape sections concerning higher-order jet bundles on Fermat type hypersurfaces, aswell as on their intersections.
Proof.
According to the criterion (50) of Proposition 6.2, it is necessary and su ffi cient to show, forall z ∈ (cid:98) X , for all [ ξ ] ∈ T hor (cid:98) V (cid:12)(cid:12)(cid:12) z and for all λ ∈ K × , that: (cid:98) ω i ,..., i l j ,..., j l (cid:0) λ z , [ λξ ] (cid:1) = λ ♥ (cid:98) ω i ,..., i l j ,..., j l (cid:0) z , [ ξ ] (cid:1) . (72) e may assume z ∈ (cid:98) U = { z (cid:44) } for instance. Now, applying Proposition 6.9, we receive: (cid:98) ω i ,..., i l j ,..., j l (cid:0) λ z , [ λξ ] (cid:1) = (cid:98) ω i ,..., i l j ,..., j l ;0 (cid:0) λ z , [ λξ ] (cid:1) = z λ − (cid:101) A i · · · (cid:101) A Ni ... ... (cid:101) A i l · · · (cid:101) A Ni l B j · · · B Nj ... ... B j l · · · B Nj l (cid:0) λ z , λξ (cid:1) . (73)For all j = · · · N , for all p = · · · l , and for all q = · · · l , recall the degree identity (52) whichshows that the entry (cid:101) A ji p = z j A ji is a homogeneous polynomial of degree:deg A ji p + = deg F i p − λ j + , and therefore satisfies: (cid:101) A ji p ( λ z ) = λ deg F ip − λ j + (cid:101) A ji p ( z ) . (74)Recalling also the notation (54), the entry B jj q is a 1-form satisfying: B jj q (cid:0) λ z , λ [ ξ ] (cid:1) = λ deg A jjq + B jj q (cid:0) z , [ ξ ] (cid:1) = λ deg F jq − λ j + B jj q (cid:0) z , [ ξ ] (cid:1) . (75)Now, let us continue to compute (73), starting by expanding the determinant: (cid:101) A i · · · (cid:101) A Ni ... ... (cid:101) A i l · · · (cid:101) A Ni l B j · · · B Nj ... ... B j l · · · B Nj l (cid:0) λ z , λξ (cid:1) = (cid:88) σ ∈ S N sign( σ ) (cid:101) A σ (1) i · · · (cid:101) A σ ( l ) i l B σ ( l + j · · · B σ ( l + l ) j l (cid:0) λ z , λξ (cid:1) . (76) With the help of the above two entry identities (74) and (75), each term in the above sum equals to:sign( σ ) (cid:101) A σ (1) i · · · (cid:101) A σ ( l ) i l B σ ( l + j · · · B σ ( l + l ) j l (cid:0) z , ξ (cid:1) ultiplied by λ ? , where: ? = l (cid:88) p = (cid:0) deg F i p − λ σ ( p ) + (cid:1) + l (cid:88) q = (cid:0) deg F j q − λ σ ( l + q ) + (cid:1) = l (cid:88) p = deg F i p + l (cid:88) q = deg F j q − (cid:16) l (cid:88) p = λ σ ( p ) + l (cid:88) q = λ σ ( l + q ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = (cid:80) l + l j = λ σ ( j ) (cid:17) + l + l (cid:124)(cid:123)(cid:122)(cid:125) = N = l (cid:88) p = deg F i p + l (cid:88) q = deg F j q − N (cid:88) j = λ j + N [Use (71) ] = ♥ + λ − , therefore (76) factors as: λ ♥ + λ − (cid:88) σ ∈ S N sign( σ ) (cid:101) A σ (1) i · · · (cid:101) A σ ( l ) i l B σ ( l + j · · · B σ ( l + l ) j l (cid:0) z , ξ (cid:1) = λ ♥ + λ − (cid:101) A i · · · (cid:101) A Ni ... ... (cid:101) A i l · · · (cid:101) A Ni l B j · · · B Nj ... ... B j l · · · B Nj l (cid:0) z , ξ (cid:1) , and thus (73) becomes: λ z ) λ − λ ♥ + λ − (cid:101) A i · · · (cid:101) A Ni ... ... (cid:101) A i l · · · (cid:101) A Ni l B j · · · B Nj ... ... B j l · · · B Nj l (cid:0) z , ξ (cid:1) = λ ♥ z λ − (cid:101) A i · · · (cid:101) A Ni ... ... (cid:101) A i l · · · (cid:101) A Ni l B j · · · B Nj ... ... B j l · · · B Nj l (cid:0) z , ξ (cid:1) = λ ♥ (cid:98) ω i ,..., i l j ,..., j l ;0 (cid:0) z , [ ξ ] (cid:1) = λ ♥ (cid:98) ω i ,..., i l j ,..., j l (cid:0) z , [ ξ ] (cid:1) , which is exactly our desired equality (72). (cid:3) Now, let K be the ( c + r + c ) × ( N +
1) matrix whose first c + r rows consist of all ( N +
1) termsin the expressions of F , . . . , F c + r in the exact order, i.e. the ( i , j )-th entry of K is: K i , j : = A j − i z λ j − j − i = ··· c + r ; j = ··· N + , (77)and whose last c rows consist of all ( N +
1) terms in the expressions of dF , . . . , dF c in the exactorder, i.e. the ( c + r + i , j )-th entry of K is: K c + r + i , j : = d (cid:0) A j − i z λ j − j − (cid:1) ( i = ··· c ; j = ··· N + . The j -th column K j of K and the j -th column C j of C are proportional: K j = C j z λ j − − j − j = ··· N + . (78) n later applications, we will use Proposition 6.10 in the case: l = c + r , l = n , and in abbreviation, dropping the upper indices, we will write these global symmetric di ff eren-tial forms ω ,..., c + rj ,..., j n as ω j ,..., j n . Since we will mainly consider the case where all coordinates arenonvanishing: z (cid:44) , . . . , z N (cid:44) , the corresponding symmetric horizontal di ff erential n -forms (cid:98) ω j ,..., j n ; j of Proposition 6.8 read, inthe set { z · · · z N (cid:44) } , as: (cid:98) ω j ,..., j n ; j = ( − j z λ j − j det (cid:0)(cid:98) C j ,..., j n ; j (cid:1) [Use (78) ] = ( − j z λ j − j (cid:16) (cid:89) (cid:54) i (cid:54) N , i (cid:44) j z λ i − i (cid:17) det (cid:0)(cid:98) K j ,..., j n ; j (cid:1) = ( − j z λ − · · · z λ N − N det (cid:0)(cid:98) K j ,..., j n ; j (cid:1) ( j = ··· N ) , (79)where (cid:98) K j ,..., j n ; j is defined as an analog of (cid:98) C j ,..., j n ; j in the obvious way.6.4. Regular twisted symmetric di ff erential forms with some vanishing coordinates. Investi-gating further the construction of symmetric di ff erential forms via Cramer’s rule, for every integer1 (cid:54) η (cid:54) n −
1, for every sequence of ascending indices:0 (cid:54) v < · · · < v η (cid:54) N , by focusing on the intersection of X with the η coordinate hyperplanes: v ,..., v η X : = X ∩ { z v = } ∩ · · · ∩ { z v η = } , we can also construct several twisted symmetric di ff erential ( n − η )-forms: Γ (cid:16) v ,..., v η X , Sym n − η Ω V (?) (cid:17) (? are twisted degrees) as follows, which will be essential ingredients towards the solution of the Debarre AmplenessConjecture.For every two positive integers l (cid:62) l with: l + l = c + r + n − η = N − η, for any two sequences of ascending positive integers:1 (cid:54) i < · · · < i l (cid:54) c + r (cid:54) j < · · · < j l (cid:54) c such that the second one is a subsequence of the first one: { j , . . . , j l } ⊂ { i , . . . , i l } , let us denote by v ,..., v η C i ,..., i l j ,..., j l the ( N − η ) × ( N − η +
1) submatrix of C determined by the ( N − η )rows i , . . . , i l , c + r + j , . . . , c + r + j l and the ( N − η +
1) columns which are complement tothe columns v + , . . . , v η +
1. Also, for every index j ∈ { , . . . , N } \ { v , . . . , v η } , let v ,..., v η (cid:98) C i ,..., i l j ,..., j l ; j enote the submatrix of v ,..., v η C i ,..., i l j ,..., j l obtained by deleting the column which is originally containedin the ( j + C . Analogously to (61)-(64), we denote: v ,..., v η W j : = { z v = } ∩ · · · ∩ { z v η = } ∩ { z j (cid:44) } ⊂ P N , whose cone is: v ,..., v η (cid:98) W j : = π − (cid:0) v ,..., v η W j (cid:1) ⊂ K N + \ { } , and we denote also: v ,..., v η U j : = v ,..., v η W j ∩ X ⊂ v ,..., v η X , whose cone is: v ,..., v η (cid:98) U j : = π − (cid:0) v ,..., v η U j (cid:1) ⊂ v ,..., v η (cid:98) X : = π − (cid:0) v ,..., v η X (cid:1) . Now we have two very analogs of Propositions 6.9 and 6.10.First, write the ( N − η +
1) remaining numbers of the set-minus: { , . . . , N } \ { v , . . . , v η } in the ascending order: r < r < · · · < r N − η . (80)It is necessary to assume that λ , . . . , λ N (cid:62) Proposition 6.11.
For all j = · · · N − η , the following ( N + − η ) a ffi ne regular symmetrichorizontal di ff erential l -forms: v ,..., v η (cid:98) ω i ,..., i l j ,..., j l ; r j : = ( − j z λ rj − r j det (cid:0) v ,..., v η (cid:98) C i ,..., i l j ,..., j l ; r j (cid:1) ∈ Γ (cid:16) v ,..., v η (cid:98) U r j , Sym l Ω hor (cid:98) V (cid:17) glue together to make a regular symmetric horizontal di ff erential l -form on v ,..., v η (cid:98) X: v ,..., v η (cid:98) ω i ,..., i l j ,..., j l ∈ Γ (cid:16) v ,..., v η (cid:98) X , Sym l Ω hor (cid:98) V (cid:17) . Proposition 6.12.
Under the assumptions and notation of the above proposition, the regular sym-metric horizontal di ff erential l -form v ,..., v η (cid:98) ω i ,..., i l j ,..., j l on v ,..., v η (cid:98) X is the image of a twisted regularsymmetric di ff erential l -form on v ,..., v η X: v ,..., v η ω i ,..., i l j ,..., j l ∈ Γ (cid:0) v ,..., v η X , Sym l Ω V ( v ,..., v η ♥ i ,..., i l j ,..., j l ) (cid:1) under the canonical injection: Γ (cid:0) v ,..., v η X , Sym l Ω V ( v ,..., v η ♥ i ,..., i l j ,..., j l ) (cid:1) (cid:44) → Γ (cid:0) v ,..., v η (cid:98) X , Sym l Ω hor (cid:98) V (cid:1) , where the twisted degree is: v ,..., v η ♥ i ,..., i l j ,..., j l : = l (cid:88) p = deg F i p + l (cid:88) q = deg F j q − (cid:16) N (cid:88) j = λ j − η (cid:88) µ = λ v µ (cid:17) + ( N − η ) + . (81) Furthermore, for all homogeneous polynomials P ∈ Γ (cid:0) P N , O P N (deg P ) (cid:1) , by multiplication, onereceives more twisted regular symmetric di ff erential l -forms:P v ,..., v η ω i ,..., i l j ,..., j l ∈ Γ (cid:0) v ,..., v η X , Sym l Ω V (deg P + v ,..., v η ♥ i ,..., i l j ,..., j l ) (cid:1) . (cid:3) n our coming applications, we will use Proposition 6.12 in the case: l = c + r , l = n − η, and in abbreviation we write these symmetric di ff erential forms v ,..., v η ω ,..., c + rj ,..., j n − η as v ,..., v η ω j ,..., j n − η .Since we will mainly consider the case when all coordinates but z v , . . . , z v η are nonvanishinig: z r (cid:44) , . . . , z r N − η (cid:44) , the corresponding symmetric horizontal di ff erential ( n − η )-forms v ,..., v η ω j ,..., j n − η of Proposition 6.11read, in the set { z r · · · z r N − η (cid:44) } , as: v ,..., v η (cid:98) ω j ,..., j l ; r j : = ( − j z λ rj − r j det (cid:0) v ,..., v η (cid:98) C i ,..., i l j ,..., j l ; r j (cid:1) [Use (78) ] = ( − j z λ rj − r j (cid:16) (cid:89) (cid:54) i (cid:54) N − η, i (cid:44) j z λ ri − r i (cid:17) det (cid:0) v ,..., v η (cid:98) K j ,..., j n − η ; r j (cid:1) = ( − j z λ r − r · · · z λ rN − η − r N − η det (cid:0) v ,..., v η (cid:98) K j ,..., j n − η ; r j (cid:1) ( j = ··· N − η ) , (82)where v ,..., v η (cid:98) K j ,..., j n − η ; r j is defined as an analog of v ,..., v η (cid:98) C j ,..., j n − η ; r j in the obvious way.The two formulas (79), (82) will enable us to e ffi ciently narrow the base loci of the obtainedsymmetric di ff erential forms, as the matrix K directly copies the original equations (cid:14) di ff erentials ofthe hypersurface polymonials F , . . . , F c + r . We will heartily appreciate such a formalism when awealth of moving coe ffi cient terms happen to tangle together.6.5. A scheme-theoretic point of view.
In future applications, we will only consider symmetricforms in coordinates . Nevertheless, in this subsection, let us reconsider the obtained symmetricforms in an algebraic way, dropping the assumption ‘algebraically-closed’ on the ambient field K .Recalling (3), (4), we may denote the projective parameter space of the c + r hypersurfaces in(51) by: P (cid:169) K = Proj K (cid:20)(cid:110) A ji ,α (cid:111) i = ··· c + rj = ··· N | α | = d i − λ j (cid:21) , so that the hypersurface coe ffi cient polynomials A ji are written as: A ji : = (cid:88) | α | = d i − λ j A ji ,α z α ( i = ··· c + r , j = ··· N ) . (83)Now, we give a scheme-theoretic explanation of Proposition 6.3, firstly by expressing (cid:98) ω j interms of a ffi ne coordinates.For every index j = · · · N , in each a ffi ne set: (cid:98) W j = { z j (cid:44) } ⊂ K N + \ { } , the c + r homogeneous hypersurface equations (51) in a ffi ne coordinates: (cid:18) z z j , . . . , (cid:98) z j z j , . . . , z N z j (cid:19) ecome: (cid:0) F i (cid:1) j = N (cid:88) k = (cid:0) A ki (cid:1) j (cid:16) z k z j (cid:17) λ k [see (55) ] = N (cid:88) k = (cid:0)(cid:101) A ki (cid:1) j (cid:16) z k z j (cid:17) λ k − , (84)where for any homogeneous polynomial P , we dehomogenize: (cid:0) P (cid:1) j : = Pz deg Pj . Di ff erentiating (84) for i = · · · c , we receive: d (cid:0) F i (cid:1) j = N (cid:88) k = B ki , j (cid:16) z k z j (cid:17) λ k − , where: B ki , j : = z k z j d (cid:0) A ki (cid:1) j + λ k (cid:0) A ki (cid:1) j d (cid:18) z k z j (cid:19) ( j , k = ··· N ) . (85)Computing z k d (cid:0) A ki (cid:1) j , we receive: z k d (cid:0) A ki (cid:1) j = z k d (cid:18) A ki z d i − λ k j (cid:19) [use (52) ][Leibniz’s rule] = z k d A ki z d i − λ k j − ( d i − λ k ) z k A ki z d i − λ k + j dz j [use (53) ] = (cid:0) B ki − λ k A ki dz k (cid:1) z d i − λ j j − ( d i − λ k ) z k A ki z d i − λ j + j dz j , therefore (85) become: B ki , j = (cid:0) B ki − λ k A ki dz k (cid:1) z d i − λ j + j − ( d i − λ k ) z k A ki z d i − λ j + j dz j + λ k (cid:0) A ki (cid:1) j d (cid:18) z k z j (cid:19) = B ki z d i − λ j + j − λ k (cid:0) A ki (cid:1) j dz k z j − ( d i − λ k ) (cid:0) A ki (cid:1) j z k z j dz j + λ k (cid:0) A ki (cid:1) j (cid:16) dz k z j − z k z j dz j (cid:17) = z d i − λ j + j B ki − d i z k z j dz j (cid:0) A ki (cid:1) j = z d i − λ j + j B ki − d i z j dz j (cid:0)(cid:101) A ki (cid:1) j . (86) ecalling the matrix C in (57), which is obtained by copying the homogeneous hypersurfaceequations F , . . . , F c + r and the di ff erentials dF , . . . , dF c , we define the matrix: (cid:0) C (cid:1) j : = (cid:0)(cid:101) A (cid:1) j · · · (cid:0)(cid:101) A N (cid:1) j ... ... (cid:0)(cid:101) A c + r (cid:1) j · · · (cid:0)(cid:101) A Nc + r (cid:1) j B , j · · · B N , j ... ... B c , j · · · B Nc , j , (87)which is obtained by copying the dehomogenized hypersurface equations (cid:0) F (cid:1) j , . . . , (cid:0) F c + r (cid:1) j andthe di ff erentials d (cid:0) F (cid:1) j , . . . , d (cid:0) F c (cid:1) j . Recalling the matrices (59), (60), in the obvious way we alsodefine (cid:0) D (cid:1) j , (cid:0)(cid:98) D k (cid:1) j as: (cid:0) D (cid:1) j : = (cid:0)(cid:101) A (cid:1) j · · · (cid:0)(cid:101) A N (cid:1) j ... ... (cid:0)(cid:101) A c + r (cid:1) j · · · (cid:0)(cid:101) A Nc + r (cid:1) j B , j · · · B N , j ... ... B n , j · · · B Nn , j , (88)and: (cid:0)(cid:98) D k (cid:1) j : = (cid:0)(cid:101) A (cid:1) j · · · (cid:91) (cid:0)(cid:101) A k (cid:1) j . . . (cid:0)(cid:101) A N (cid:1) j ... ... (cid:0)(cid:101) A c + r (cid:1) j · · · (cid:91) (cid:0)(cid:101) A kc + r (cid:1) j . . . (cid:0)(cid:101) A Nc + r (cid:1) j B , j · · · (cid:100) B k , j . . . B N , j ... ... B n , j · · · (cid:100) B kn , j . . . B Nn , j ( k = ··· N ) . (89)Recalling (cid:98) ω j of Proposition 6.3, now thanks to (86), we have the following nice: Observation 6.13.
For every j = · · · N, one has the identity: (cid:98) ω j = ( − j z λ j − j det (cid:0)(cid:98) D j (cid:1) = ( − j z −♥ j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) , (90) where for the moment ♥ is defined in (71) for ω ,..., c + r ,..., c : ♥ : = c + r (cid:88) p = d p + n (cid:88) q = d q − N (cid:88) j = λ j + N + . The proof is much the same as that of Proposition 6.10, hence we omit it here. (cid:3) ow, let pr , pr be the two canonical projections: P (cid:169) K × K P N K pr (cid:123) (cid:123) pr (cid:35) (cid:35) P (cid:169) K P N K . Then thanks to the formula (90), we may view (cid:98) ω j as a section of the twisted sheaf: Sym n Ω P (cid:169) K × K P N K / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) ⊗ pr ∗ O P N K ( ♥ )over the pullback: pr − ( W j ) ⊂ P (cid:169) K × K P N K of the canonical a ffi ne scheme W j : = D ( z j ) ⊂ P N K . Using the same notation as (5), (6), recalling (51), (83), we now introduce the two subschemes: X ⊂ V ⊂ P (cid:169) K × K P N K , where X is defined by ‘all’ the c + r bihomogeneous polynomials: X : = V (cid:16) N (cid:88) j = A j z λ j j , . . . , N (cid:88) j = A jc z λ j j , N (cid:88) j = A jc + z λ j j , . . . , N (cid:88) j = A jc + r z λ j j (cid:17) , and where V is defined by the ‘first’ c bihomogeneous polynomials: V : = V (cid:16) N (cid:88) j = A j z λ j j , . . . , N (cid:88) j = A jc z λ j j (cid:17) . Now, we may view each entry of the matrix (87) as a section in: Γ (cid:16) X ∩ pr − ( W j ) , Sym • Ω V / P (cid:169) K ⊗ pr ∗ O P (cid:169) K (1) (cid:17) , where the symmetric degrees are 0 for the first c + r rows and 1 for the last n rows. Noting that the N + C , · · · , C N of this matrix satisfy the relation: N (cid:88) k = C k z λ k − k z λ k − j = , in particular, so do the columns of the submatrix (88). Now, recalling the submatrices (89) of (88),an application of Cramer’s rule (Theorem 6.6) yields:( − k det (cid:16)(cid:0)(cid:98) D k (cid:1) j (cid:17) z λ k − k z λ k − j = ( − k det (cid:16)(cid:0)(cid:98) D k (cid:1) j (cid:17) z λ k − k z λ k − j ∈ Γ (cid:16) X ∩ pr − ( W j ) , Sym n Ω V / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) (cid:17) ( k = ··· N ) . (91)Now, recalling (90), we may interpret Proposition 6.7 as follows. First, for j = · · · N , we vieweach: (cid:98) ω j = ( − j z −♥ j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) s a section in: Γ (cid:16) X ∩ pr − ( W j ) , Sym n Ω V / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) ⊗ pr ∗ O P N K ( ♥ ) (cid:17) . Then, thanks to an observation below, for every di ff erent indices j < j , over the open set: X ∩ pr − ( W j ∩ W j ) ⊂ X , the twisted sheaf: Sym n Ω V / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) ⊗ pr ∗ O P N K ( ♥ )has the two coinciding sections: (cid:98) ω j = ( − j z −♥ j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) [Observation 6.13][ use (91) ] = ( − j z −♥ j z λ j − j z λ j − j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) [ Observation 6.14 below ] = ( − j z −♥ j z λ j − j z λ j − j z ♥ + λ j − j z ♥ + λ j − j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) = ( − j z −♥ j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) = (cid:98) ω j . Thus, the N + (cid:98) ω , . . . , (cid:98) ω N glue together to make a global section: (cid:98) ω ∈ Γ (cid:16) X , Sym n Ω V / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) ⊗ pr ∗ O P N K ( ♥ ) (cid:17) . Observation 6.14.
For all distinct indices (cid:54) j , j (cid:54) N, one has the transition identities: det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) = z ♥ + λ j − j z ♥ + λ j − j det (cid:16)(cid:0)(cid:98) D j (cid:1) j (cid:17) . The proof is but elementary computations, so we omit it here. (cid:3)
Next, repeating the same reasoning, using the obvious notation, we interpret Propositions 6.9and 6.10 as:
Proposition 6.15.
Each of the following N + symmetric forms: (cid:98) ω i ,..., i l j ,..., j l ; j = ( − j z λ j − j det (cid:16)(cid:98) C i ,..., i l j ,..., j l ; j (cid:17) = ( − j z −♥ j det (cid:18) (cid:16)(cid:98) C i ,..., i l j ,..., j l ; j (cid:17) j (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) guess what? (cid:19) ( j = ··· N ) can be viewed as a section of: Γ (cid:16) pr − ( W j ) , Sym n Ω P (cid:169) K × K P N K / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) ⊗ pr ∗ O P N K ( ♥ ) (cid:17) , with the twisted degree: ♥ : = l (cid:88) p = deg F i p + l (cid:88) q = deg F j q − N (cid:88) j = λ j + N + . oreover, restricting on X , they glue together to make a global section: (cid:98) ω i ,..., i l j ,..., j l ∈ Γ (cid:16) X , Sym l Ω V / P (cid:169) K ⊗ pr ∗ O P (cid:169) K ( N ) ⊗ pr ∗ O P N K ( ♥ ) (cid:17) . (cid:3) We may view Propositions 6.11 and 6.12 in a similar way.7.
Moving Coe ffi cients Method Algorithm.
As explained in Subsection 4.3, we wish to construct su ffi ciently many nega-tively twisted symmetric di ff erential forms, and for this purpose we investigate the moving coe ffi -cients method as follows. We will be concerned only with the central cases that all c + r hypersur-faces are of the approximating big degrees d + (cid:15) , . . . , d + (cid:15) c + r , where (cid:15) , . . . , (cid:15) c + r are some givenpositive integers negligible compared with the large integer d (cid:29) ffi cients method while avoiding unnecessary com-plexity (see Section 12), we first consider the following c + r cumbersome homogeneous polyno-mials F , . . . , F c + r , each being the sum of a dominant Fermat-type polynomial plus an ‘army’ of moving coe ffi cient terms: F i = N (cid:88) j = A ji z dj + N (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) N l (cid:88) k = M j ,..., j l ; j k i z µ l , k j · · · (cid:99) z µ l , k j k · · · z µ l , k j l z d − l µ l , k j k , (92)where all coe ffi cients A • i , M • ; • i ∈ K [ z , . . . , z N ] are some degree (cid:15) i (cid:62) µ l , k (cid:62) d (cid:29) Algorithm , which isdesigned to make all the twisted symmetric di ff erential forms obtained later have negative twisteddegrees .The procedure is to first construct µ l , k in a lexicographic order with respect to indices ( l , k ), for l = c + r + · · · N , k = · · · l , along with a set of positive integers δ l .Recall the integer (cid:114) (cid:62) δ c + r + (cid:62) max { (cid:15) , . . . , (cid:15) c + r } . (93)For every integer l = c + r + · · · N , in this step, we begin with choosing µ l , that satisfies: µ l , (cid:62) l δ l + l ( δ c + r + + + + ( l − c − r ) (cid:114) , (94)then inductively we choose µ l , k with: µ l , k (cid:62) k − (cid:88) j = l µ l , j + ( l − k ) δ l + l ( δ c + r + + + + ( l − c − r ) (cid:114) ( k = ··· l ) . (95)If l < N , we end this step by setting: δ l + : = l µ l , l (96)as the starting point for the next step l +
1. At the end l = N , we demand the integer d (cid:29) d (cid:62) ( N + µ N , N . (97)Roughly speaking, the Algorithm above is designed for the following three properties. (i) For every integer l = c + r + · · · N , in this step, µ l , • ( • = · · · l ) grows so drastically thatthe former ones are negligible compared with the later ones: µ l , (cid:28) µ l , (cid:28) · · · (cid:28) µ l , l . (98) ii) For all integer pairs ( l , l ) with c + r + (cid:54) l < l (cid:54) N , all the integers µ l , • chosen in theformer step l are negligible compared with the integers µ l , • chosen in the later step l : µ l , • (cid:28) µ l , • ( ∀ (cid:54) • (cid:54) l ; 0 (cid:54) • (cid:54) l ) . (99) (iii) All integers µ l , k are negligible compared with the integer d : µ l , k (cid:28) d ( ∀ c + r + (cid:54) l (cid:54) N ; 0 (cid:54) k (cid:54) l ) . (100)7.2. Global moving coe ffi cients method. First, for all i = · · · c + r , we write the polynomial F i by extracting the terms for which l = N : F i = N (cid:88) j = A ji z dj + N − (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) N l (cid:88) k = M j ,..., j l ; j k i z µ l , k j · · · (cid:99) z µ l , k j k · · · z µ l , k j l z d − l µ l , k j k ++ N (cid:88) k = M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k , (101)and now this second line consists of exactly all the moving coe ffi cient terms which associate to allvariables z , . . . , z N , namely of the form M • ; • i z • · · · z • N .To simplify the structure of the first line, associating each term in the second sums: M j ,..., j l ; j k i z µ l , k j · · · (cid:99) z µ l , k j k · · · z µ l , k j l z d − l µ l , k j k with the ‘corresponding’ term in the first sum: A j k i z dj k , and noting a priori the inequalities guaranteed by the Algorithm: d − l µ l , k (cid:62) d − ( N − µ N − , N − (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = δ N [By (96) ] ( ∀ c + r + (cid:54) l (cid:54) N −
1; 0 (cid:54) k (cid:54) l ) , we rewrite the F i as: F i = N (cid:88) j = C ji z d − δ N j + N (cid:88) k = M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k , (102)where the homogeneous polynomials C ji are uniquely determined by gathering: C ji z d − δ N j = A ji z dj + N − (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) Nj k = j for some 0 (cid:54) k (cid:54) l M j ,..., j l ; j k i z µ l , k j · · · (cid:99) z µ l , k j k · · · z µ l , k j l z d − l µ l , k j k , (103)namely, after dividing out the common factor z d − δ N j of both sides above: C ji : = A ji z δ N j + N − (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) Nj k = j for some 0 (cid:54) k (cid:54) l M j ,..., j l ; j k i z µ l , k j · · · (cid:99) z µ l , k j k · · · z µ l , k j l z δ N − l µ l , k j k . (104)Next, we have two ways to manipulate the ( N +
1) remaining moving coe ffi cient terms in (102): M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k ( k = ··· N ) , in order to ensure the negativity of the symmetric di ff erential forms to be obtained later. he first kind of manipulations are, for every chosen index ν = · · · N , to associate all these( N +
1) moving coe ffi cient terms: N (cid:88) k = M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k with the term C ν i z d − δ N ν by rewriting F i in (102) as: F i = N (cid:88) j = j (cid:44) ν C ji z d − δ N j + T ν i z µ N , ν , (105)where T ν i is the homogeneous polynomial uniquely determined by solving: T ν i z µ N , ν = C ν i z d − δ N ν + N (cid:88) k = M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k ; (106)in fact, guided by properties (98), (99), (100), our algorithm a priori implies: µ N , (cid:54) d − δ N , µ N , k , d − N µ N , k ( k = ··· N ) , thus the right hand side of (106) has a common factor z µ N , ν .The second kind of manipulations are, for every chosen integer τ = · · · N −
1, for every chosenindex ρ = τ + · · · N , to associate each of the first ( τ +
1) moving coe ffi cient terms: M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k ( k = ··· τ ) with the corresponding terms C ki z d − δ N k and to associate the remaining ( N − τ ) moving coe ffi cientterms: N (cid:88) j = τ + M ,..., N ; ji z µ N , j · · · (cid:100) z µ N , j j · · · z µ N , j N z d − N µ N , j j with the term C ρ i z d − δ N ρ by rewriting F i as: F i = τ (cid:88) k = E ki z d − N µ N , k k + N (cid:88) j = τ + j (cid:44) ρ C ji z d − δ N j + P τ,ρ i z µ N ,τ + ρ , (107)where E ki and P τ,ρ i are the homogeneous polynomials uniquely determined by solving: E ki z d − N µ N , k k = C ki z d − δ N k + M ,..., N ; ki z µ N , k · · · (cid:100) z µ N , k k · · · z µ N , k N z d − N µ N , k k ( k = ··· τ ) , P τ,ρ i z µ N ,τ + ρ = C ρ i z d − δ N ρ + N (cid:88) j = τ + M ,..., N ; ji z µ N , j · · · (cid:100) z µ N , j j · · · z µ N , j N z d − N µ N , j j , (108)which is direct by the inequalities listed below granted by the Algorithm: d − N µ N , k (cid:54) d − δ N ( k = ··· τ ) ,µ N ,τ + (cid:54) µ N , j ,µ N ,τ + (cid:54) d − N µ N , j ( j = τ + ··· N ) . (109) ow thanks to the above two kinds of manipulations (105), (107), applying Proposition 6.8,6.10, we receive the corresponding twisted symmetric di ff erential forms with negative degrees asfollows.Firstly, for every index ν = · · · N , applying Proposition 6.8, 6.10 with respect to the first kindof manipulation (105) on the hypersurface polynomial equations F , . . . , F c + r , for every n -tuple1 (cid:54) j < · · · < j n (cid:54) c , we receive a twisted symmetric di ff erential n -form: φ ν j ,..., j n ∈ Γ (cid:0) X , Sym n Ω V ( ♥ ν j ,..., j n ) (cid:1) , (110)whose twisted degree ♥ ν j ,..., j n , according to the formula (71), is negative: [Use deg F i = d + (cid:15) i (cid:54) d + δ c + r + ] ♥ ν j ,..., j n (cid:54) N ( d + δ c + r + ) − (cid:2) N ( d − δ N ) + µ N , (cid:3) + N + = N δ N + N ( δ c + r + + + − µ N , [Use (94) for l = N ] (cid:54) − n (cid:114) . Secondly, for every integer τ = · · · N −
1, for every index ρ = τ + · · · N , applying Proposi-tion 6.8, 6.10 with respect to the second kind of manipulation (107) on the hypersurface polynomialequations F , . . . , F c + r , for every n -tuple 1 (cid:54) j < · · · < j n (cid:54) c , we receive a twisted symmetricdi ff erential n -form: ψ τ,ρ j ,..., j n ∈ Γ (cid:0) X , Sym n Ω V ( ♥ τ,ρ j ,..., j n ) (cid:1) , (111)whose twisted degree ♥ τ,ρ j ,..., j n , according to the formula (71), is negative too: ♥ τ,ρ j ,..., j n (cid:54) N ( d + δ c + r + ) − τ (cid:88) k = ( d − N µ N , k ) − ( N − τ −
1) ( d − δ N ) − µ N ,τ + + N + = τ (cid:88) k = N µ N , k + ( N − τ − δ N + N ( δ c + r + + + − µ N ,τ + (cid:54) − n (cid:114) [use (95) for l = N , k = τ + ] . Moving coe ffi cients method with some vanishing coordinates. To investigate further themoving coe ffi cients method, for all integers 1 (cid:54) η (cid:54) n −
1, for every sequence of ascending indices: 0 (cid:54) v < · · · < v η (cid:54) N , take the intersection of X with the η coordinate hyperplanes: v ,..., v η X : = X ∩ { z v = } ∩ · · · ∩ { z v η = } . Applying Proposition 6.12, in order to obtain more symmetric di ff erential ( n − η )-forms havingnegative twisted degree, we carry on manipulations as follows, which are much the same as before.First, write the ( N − η +
1) remaining numbers of the set-minus: { , . . . , N } \ { v , . . . , v η } in the ascending order: r < · · · < r N − η . Note that in Proposition 6.12, the coe ffi cient terms associated with the vanishing variables z v , . . . , z v η play no role, therefore we decompose F i into two parts. The first part (the first two lines below) s a very analog of (101) involving only the variables z r , . . . , z r N − η , while the second part (the thirdline) collects all the residue terms involving at least one of the vanishing coordinates z v , . . . , z v η : F i = N − η (cid:88) j = A r j i z dr j + N − η − (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) N − η l (cid:88) k = M r j ,..., r jl ; r jk i z µ l , k r j · · · (cid:99) z µ l , k r jk · · · z µ l , k r jl z d − l µ l , k r jk ++ N − η (cid:88) k = M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k ++ (Residue Terms) v ,..., v η i (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) negligible in the coming applications . (112)Since every power associated with the vanishing variables z v , . . . , z v v η is (cid:62) v ,..., v η i lie in the ideal:( z v , . . . , z v η ) ⊂ K [ z , . . . , z N ] . Moreover, using for instance the lexicographic order, we can write them as:(Residue Terms) v ,..., v η i = η (cid:88) j = R v ,..., v η ; v j i z v j , (113)where R v ,..., v η ; v j i are the homogeneous polynomials uniquely determined by solving: R v ,..., v η ; v j i z v j = A v j i z dv j + N (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) N min (cid:0) { j ,..., j l }\{ v ,..., v η } (cid:1) = v j l (cid:88) k = M j ,..., j l ; j k i z µ l , k j · · · (cid:99) z µ l , k j k · · · z µ l , k j l z d − l µ l , k j k . Observing that the first two lines of (112) have exactly the same structure as (101), by mimickingthe manipulation of rewriting (101) as (102), we can rewrite the first two lines of (112) as: N − η (cid:88) j = v ,..., v η C r j i z d − δ N − η r j + N − η (cid:88) k = M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k , (114)where the integer δ N − η was defined in (96) for l = N − η − δ N − η = ( N − η − µ N − η − , N − η − , and where the homogeneous polynomials v ,..., v η C r j i are obtained in the same way as C ji in (104): v ,..., v η C r j i : = A r j i z δ N − η j + N − η − (cid:88) l = c + r + (cid:88) (cid:54) j < ··· < j l (cid:54) N − η j k = j for some 0 (cid:54) k (cid:54) l M r j ,..., r jl ; r jk i z µ l , k r j · · · (cid:99) z µ l , k r jk · · · z µ l , k r jl z ( N − η − µ N − η − , N − η − − l µ l , k r jk . Now substituting (114), (113) into the equation (112), we rewrite F i as: F i = N − η (cid:88) j = v ,..., v η C r j i z d − δ N − η r j + N − η (cid:88) k = M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k ++ η (cid:88) j = R v ,..., v η ; v j i z v j . (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) negligible in the coming applications (115) ow, noting that the first line of F i in (115) has exactly the same structure as (102), we repeatthe two kinds of manipulations, as briefly summarized below.The first kind of manipulations are, for every chosen index ν = · · · N − η , to associate all these( N + − η ) moving coe ffi cient terms: N − η (cid:88) k = M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k with the term v ,..., v η C r ν i z d − δ N − η r ν by rewriting (114) as: N − η (cid:88) j = j (cid:44) ν v ,..., v η C r j i z d − δ N − η r j + v ,..., v η T r ν i z µ N − η, r ν , (116)where v ,..., v η T r ν i is the homogeneous polynomial uniquely determined by solving: v ,..., v η T r ν i z µ N − η, r ν = v ,..., v η C r ν i z d − δ N − η r ν + N − η (cid:88) k = M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k . (117) The second kind of manipulations are, for every integer τ = · · · N − η −
1, for every index ρ = τ + · · · N − η , to associate each of the first ( τ +
1) moving coe ffi cient terms: M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k ( k = ··· τ ) with the corresponding term v ,..., v η C r k i z d − δ N − η r k and to associate the remaining ( N − η − τ ) movingcoe ffi cient terms: N − η (cid:88) j = τ + M r ,..., r N − η ; r j i z µ N − η, j r · · · (cid:91) z µ N − η, j r j · · · z µ N − η, j r N − η z d − ( N − η ) µ N − η, j r j , with the term v ,..., v η C r ρ i z d − δ N − η r ρ by rewriting (114) as: τ (cid:88) k = v ,..., v η E r k i z d − ( N − η ) µ N − η, k r k + N − η (cid:88) j = τ + j (cid:44) ρ v ,..., v η C r j i z d − δ N − η r j + v ,..., v η P r τ , r ρ i z µ N − η,τ + r ρ , (118)where v ,..., v η E r k i and v ,..., v η P r τ , r ρ i are the homogeneous polynomials uniquely determined by solving: v ,..., v η E r k i z d − ( N − η ) µ N − η, k r k = v ,..., v η C r k i z d − δ N − η r k + M r ,..., r N − η ; r k i z µ N − η, k r · · · (cid:91) z µ N − η, k r k · · · z µ N − η, k r N − η z d − ( N − η ) µ N − η, k r k , v ,..., v η P r τ , r ρ i z µ N − η,τ + r ρ = v ,..., v η C r ρ i z d − δ N − η r ρ + N − η (cid:88) j = τ + M r ,..., r N − η ; r j i z µ N − η, j r · · · (cid:91) z µ N − η, j r j · · · z µ N − η, j r N − η z d − ( N − η ) µ N − η, j r j , (119) which is possible by the Algorithm in subsection 7.1.To summarize, taking the two forms (116), (118) of the first line of (115) into account, we canrewrite F i in the following two ways. The first one is: F i = N − η (cid:88) j = j (cid:44) ν v ,..., v η C r j i z d − δ N − η r j + v ,..., v η T r ν i z µ N − η, r ν + η (cid:88) j = R v ,..., v η ; v j i z v j (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) negligible in our coming applications , (120) nd the second one is: F i = τ (cid:88) k = v ,..., v η E r k i z d − ( N − η ) µ N − η, k r k + N − η (cid:88) j = τ + j (cid:44) ρ v ,..., v η C r j i z d − δ N − η r j + v ,..., v η P r τ , r ρ i z µ N − η,τ + r ρ + η (cid:88) j = R v ,..., v η ; v j i z v j (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) negligible in our coming applications . (121) Firstly, applying Proposition 6.12 to (120), for every ( n − η )-tuple:1 (cid:54) j < · · · < j n − η (cid:54) c , we receive a twisted symmetric di ff erential ( n − η )-form: v ,..., v η φ ν j ,..., j n − η ∈ Γ (cid:0) v ,..., v η X , Sym n − η Ω V ( v ,..., v η ♥ ν j ,..., j n − η ) (cid:1) , (122)whose twisted degree v ,..., v η ♥ ν j ,..., j n − η , according to the formula (81), is negative: v ,..., v η ♥ ν j ,..., j n − η (cid:54) ( N − η ) ( d + δ c + r + ) − (cid:2) ( N − η ) ( d − δ N − η ) + µ N − η, (cid:3) + ( N − η ) + = ( N − η ) δ N − η + ( N − η ) ( δ c + r + + + − µ N − η, [use (94) for l = N − η ] (cid:54) − ( n − η ) (cid:114) . Secondly, applying Proposition 6.12 to (121), for every ( n − η )-tuple:1 (cid:54) j < · · · < j n − η (cid:54) c , we receive a twisted symmetric di ff erential ( n − η )-form: v ,..., v η ψ τ,ρ j ,..., j n − η ∈ Γ (cid:0) v ,..., v η X , Sym n − η Ω V ( v ,..., v η ♥ τ,ρ j ,..., j n − η ) (cid:1) , (123)whose twisted degree v ,..., v η ♥ τ,ρ j ,..., j n − η , according to the formula (81), is negative also: v ,..., v η ♥ τ,ρ j ,..., j n − η (cid:54) ( N − η ) ( d + δ c + r + ) − τ (cid:88) k = ( d − ( N − η ) µ N − η, k ) − ( N − η − τ −
1) ( d − δ N − η ) −− µ N − η,τ + + N − η + = τ (cid:88) k = ( N − η ) µ N − η, k + ( N − η − τ − δ N − η + ( N − η ) ( δ c + r + + + − µ N − η,τ + (cid:54) − ( n − η ) (cid:114) [use (95) for l = N − η , k = τ + ] . Basic technical preparations
Fibre dimension estimates.
The first theorem below coincides with our geometric intuition,and one possible proof is mainly based on Complex Implicit Function Theorem. Here, we give ashort proof by the same method as [59, p. 76, Theorem 7].
Theorem 8.1. (Analytic Fibre Dimension Estimate)
Let X , Y be two complex spaces and letf : X → Y be a regular map. Then the maximum fibre dimension is bounded from below by thedimension of the source space X minus the dimension of the target space Y: max y ∈ Y dim C f − ( y ) (cid:62) dim C X − dim C Y . Equivalently: dim C X (cid:54) dim C Y + max y ∈ Y dim C f − ( y ) . (124) roof. For every point x ∈ X , let f ( x ) = : z ∈ Y , and denote the germ dimension of Y at this pointby: d z : = dim C ( Y , z ) . Then we can find holomorphic function germs g , . . . , g d z ∈ O Y , z vanishing at z such that:( Y , z ) ∩ { g = · · · = g d z = } = { z } . Pulling back by the holomorphic map f , we therefore realize:( X , x ) ∩ { g ◦ f = · · · = g d z ◦ f = } = (cid:0) f − ( z ) , x (cid:1) . Now, counting the germ dimension, we receive the estimate: dim C (cid:0) f − ( z ) , x (cid:1) (cid:62) dim C ( X , x ) − d z = dim C ( X , x ) − dim C ( Y , z ) , hence: dim C ( X , x ) (cid:54) dim C ( Y , z ) + dim C (cid:0) f − ( z ) , x (cid:1) (cid:54) dim C Y + max y ∈ Y dim C f − ( y ) . Finally, let x ∈ X vary in the above estimate, thanks to: dim C X = max x ∈ X dim C ( X , x ) , we receive the desired estimate (124). (cid:3) With the same proof (cf. [54, p. 169, Proposition 12.30; p. 140, Corollary 10.27]), here is analgebraic version of the analytic fibre dimension estimate above, for every algebraically closed field K and for the category of K -varieties in the classical sense ([37, §1.3, p. 15]), where dimension isdefined to be the Krull dimension ([37, §1.1, p. 6]). Theorem 8.2 ( Algebraic Fibre Dimension Estimate).
Let X , Y be two K -varieties, and let f : X → Y be a morphism. Then the dimension of the source variety X is bounded from above by the sumof the dimension of the target variety Y plus the maximum fibre dimension: dim X (cid:54) dim Y + max y ∈ Y dim f − ( y ) . (125)In our future applications, f will always be surjective, so one may also refer to [59, p. 76,Theorem 7]. The above theorem will prove fundamental in estimating every base locus involvedin this paper. Corollary 8.3.
Let X , Y be two K -varieties, and let f : X → Y be a morphism such that every fibresatisfies the dimension estimate: dim f − ( y ) (cid:54) dim X − dim Y ( ∀ y ∈ Y ) . Then for every subvariety Z ⊂ Y, its inverse image:f − ( Z ) ⊂ Xsatisfies the transferred codimension estimate: codim f − ( Z ) (cid:62) codim Z . (cid:3) .2. Matrix-rank estimates.
This subsection recalls some elementary rank estimates in linearalgebra.
Lemma 8.4.
Let K be a field and let W be a finite-dimensional K -vector space generated by a setof vectors B . Then every subset B ⊂ B that consists of K -linearly independent vectors can beextended to a bigger subset B ⊂ B which forms a basis of W. (cid:3) Lemma 8.5.
Let K be a field, and let V be a K -vector space. For all positive integers e , k , l (cid:62) with k (cid:62) l, let v , . . . , v e , v e + , . . . , v e + k be ( e + k ) vectors such that: (i) v , . . . , v e are K -linearly independent; (ii) for every sequence of l ascending indices between e + and e + k:e + (cid:54) i < · · · < i l (cid:54) e + k , there holds the rank inequality: rank K { v , . . . , v e , v i , . . . , v i l } (cid:54) e + l − . Then there holds the rank estimate: rank K { v , . . . , v e , v e + , . . . , v e + k } (cid:54) e + l − . Proof.
Assume on the contrary that: rank K { v , . . . , v e , v e + , . . . , v e + k } = : e + l (cid:62) e + l , that is, l (cid:62) l .Now applying the above lemma to: W = Span K { v , . . . , v e , v e + , . . . , v e + k } (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = B , B = { v , . . . , v e } , we receive a certain basis of V : B = { v , . . . , v e , v i , . . . , v i l } . In particular, as l (cid:62) l , the first ( e + l ) vectors in B are K -linearly independent: rank K { v , . . . , v e , v i , . . . , v i l } = e + l , which contradicts condition (ii) . (cid:3) Let K be a field, and let p , q , e , l be positive integers with: min { p , q } (cid:62) e + l . Let M ∈ Mat p × q ( K ) be a p × q matrix. For all sequences of ascending indices :1 (cid:54) i < · · · < i k (cid:54) p , let us denote by M i ,..., i k the k × q submatrix of M that consists of the rows i , . . . , i k , and for allsequences of ascending indices: 1 (cid:54) j < · · · < j l (cid:54) q , let us denote by M j ,..., j l i ,..., i k the k × l submatrix of M i ,..., i k that consists of the columns j , . . . , j l . emma 8.6. If the first e rows of the matrix M are of full rank: rank K M ,..., e = e , and if all the ( e + l ) × ( e + l ) submatrices always selecting the first e rows of M are degenerate: rank K M j ,..., j e + l ,..., e , i ,..., i l (cid:54) e + l − ( ∀ e + (cid:54) i < ··· < i l (cid:54) p ; 1 (cid:54) j < ··· < j e + l (cid:54) q ) , (126) then there holds the rank estimate: rank K M (cid:54) e + l − Proof.
For every fixed sequence of ascending indices: e + (cid:54) i < · · · < i l (cid:54) p , the rank inequalities (126) yields: rank K M ,..., e , i ,..., i l (cid:54) e + l − M , we conclude the desired rankestimate. (cid:3) Lemma 8.7.
Let K be a field and let e , m be positive integers. Let H ∈ Mat e × m ( K ) be an e × mmatrix with entries in K such that the sum of all m columns of H vanishes: H + · · · + H m = , (127) where we denote by H i the i-th column of H . Then for every integer j = · · · m, the e × m submatrix (cid:98) H j of H obtained by omitting the j-th column still has the same rank: rank K (cid:98) H j = rank K H . Proof.
Note that (127) yields: H j = − ( H + · · · + H j − + H j + + · · · + H m ) , therefore H j lies in the K -linear space generated by the columns of the matrix (cid:98) H j , thus we receive: Span K { H , . . . , (cid:98) H j , . . . , H m } = Span K { H , . . . , H m } . Taking the dimension of both sides, we receive the desired rank equality. (cid:3)
Classical codimension formulas.
In an algebraically closed field K , for all positive integers p , q (cid:62)
1, denote by:
Mat p × q ( K ) = K p × q the space of all p × q matrices with entries in K . For every integer 0 (cid:54) (cid:96) (cid:54) max { p , q } , we have aclassical formula (cf. [31, p. 247 exercise 10.10, and the proof in p. 733]) for the codimension ofthe subvariety: Σ p , q (cid:96) ⊂ Mat p × q ( K )which consists of all matrices with rank (cid:54) (cid:96) . Lemma 8.8.
There holds the codimension formula: codim Σ p , q (cid:96) = max (cid:8) ( p − (cid:96) ) ( q − (cid:96) ) , (cid:9) . (cid:3) In applications, we will use the following two direct consequences. orollary 8.9. For every integer (cid:54) (cid:96) (cid:54) max { p , q − } , the codimension of the subvariety: Σ p , q (cid:96) ⊂ Σ p , q (cid:96) , which consists of matrices whose sum of all the columns vanish, is: codim Σ p , q (cid:96) = max (cid:8) ( p − (cid:96) ) ( q − − (cid:96) ) , (cid:9) + p . (cid:3) Proof.
Since every matrix in Σ p , q (cid:96) is uniquely determined by the first ( q −
1) columns, thanks toLemma 8.7, the projection morphism into the first ( q −
1) columns: π : Σ p , q (cid:96) −→ Σ p , q − (cid:96) is an isomorphism. Remembering that: dim Σ p , q (cid:96) = dim Σ p , q − (cid:96) + p , now a direct application of the preceding lemma finishes the proof. (cid:3) Surjectivity of evaluation maps.
Given a field K , for all positive integers N (cid:62)
1, denote thea ffi ne coordinate ring of K N + by: A ( K N + ) : = K [ z , . . . , z N ] . For all positive integers λ (cid:62)
1, also denote by: A λ ( K N + ) ⊂ A ( K N + )the K -linear space spanned by all the degree λ homogeneous polynomials: A λ ( K N + ) : = ⊕ α + ··· + α N = λα ,...,α N (cid:62) K · z λ · · · z λ N N (cid:27) K ( N + λ N ) . For every point z ∈ K N + , denote by v z the K -linear evaluation map: v z : A ( K N + ) −→ K f (cid:55)−→ f ( z ) , and for every tangent vector ξ ∈ T z K N + (cid:27) K N + , denote by d z ( ξ ) the K -linear di ff erential evaluationmap: d z ( ξ ) : A ( K N + ) −→ K f (cid:55)−→ d f (cid:12)(cid:12)(cid:12) z ( ξ ) . For every polynomial g ∈ A ( K N + ), for every point z ∈ K N + , denote by ( g · v ) z the K -linearevaluation map: ( g · v ) z : A ( K N + ) −→ K f (cid:55)−→ ( g f )( z ) , and for every tangent vector ξ ∈ T z K N + (cid:27) K N + , denote by d z ( g · )( ξ ) the K -linear di ff erentialevaluation map: d z ( g · )( ξ ) : A ( K N + ) −→ K f (cid:55)−→ d ( g f ) (cid:12)(cid:12)(cid:12) z ( ξ ) . The following Lemma was obtained by Brotbek in another a ffi ne coordinates version [7, p. 36,Proof of Claim 3]. emma 8.10. For all positive integers λ (cid:62) , at every nonzero point z ∈ K N + \ { } , for everytangent vector ξ ∈ T z K N + (cid:27) K N + which does not lie in the line of z: ξ (cid:60) K · z , restricting on the subspace: A λ ( K N + ) ⊂ A ( K N + ) , the evaluation maps v z and d z ( ξ ) are K -linearly independent. In other words, the map: (cid:32) v z d z ( ξ ) (cid:33) : A λ ( K N + ) −→ K is surjective.Proof. Step 1. For the case λ =
1, this lemma is evident. In fact, now every polynomials (cid:96) ∈ A ( K N + ) can be viewed as, by evaluating (cid:96) ( z ) at every point z ∈ K N + , a K -linear form: (cid:96) ∈ (cid:0) K N + (cid:1) ∨ , thus there is a canonical K -linear isomorphism: A ( K N + ) (cid:27) (cid:0) K N + (cid:1) ∨ . Moreover, it is easy to see: d (cid:96) (cid:12)(cid:12)(cid:12) z ( ξ ) = (cid:96) ( ξ ) . Since z , ξ ∈ K N + are K -linearly independent, now recalling the Riesz Representation Theorem inlinear algebra: K N + (cid:27) (cid:16)(cid:0) K N + (cid:1) ∨ (cid:17) ∨ , (128)we conclude the claim. Step 2.
For the general case λ (cid:62)
2, first, we choose a degree ( λ −
1) homogeneous polynomial g ∈ A λ − ( K N + ) with g ( z ) (cid:44) z λ − , . . . , z λ − N succeeds), and then we claim,restricting on the K -linear subspace obtained by multiplying A ( K N + ) with g : g · A ( K N + ) ⊂ A λ ( K N + ) , that the evaluation maps v z and d z ( ξ ) are K -linearly independent.In fact, for all f ∈ A ( K N + ), we have:( g · v ) z ( f ) = ( g f )( z ) = g ( z ) f ( z ) = g ( z ) v z ( f ) , and by Leibniz’s rule: d z ( g · )( ξ ) ( f ) = d ( g f ) (cid:12)(cid:12)(cid:12) z ( ξ ) = g ( z ) d f (cid:12)(cid:12)(cid:12) z ( ξ ) + f ( z ) d g (cid:12)(cid:12)(cid:12) z ( ξ ) = g ( z ) d z ( ξ ) ( f ) + d g (cid:12)(cid:12)(cid:12) z ( ξ ) v z ( f ) , in other words: (cid:32) ( g · v ) z d z ( g · )( ξ ) (cid:33) = (cid:32) g ( z ) 0 d g (cid:12)(cid:12)(cid:12) z ( ξ ) g ( z ) (cid:33)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) invertible, since g ( z ) (cid:44) (cid:32) v z d z ( ξ ) (cid:33) . (129) ow, restricting (129) on the K -linear subspace: A ( K N + ) ⊂ A ( K N + ) , and recalling the result of Step 1 that the evaluation maps v z , d z ( ξ ) are K -linearly independent, weimmediately see that the evaluation maps ( g · v ) z , d z ( g · )( ξ ) are K -linearly independent too. In otherwords, restricting on the K -linear subspace: g · A ( K N + ) ⊂ A λ ( K N + ) , the evaluation maps v z , d z ( ξ ) are K -linearly independent. (cid:3) Lemma 8.11.
For all positive integers λ (cid:62) , for all polynomials g ∈ A ( K N + ) , at every nonzeropoint z ∈ K N + \ { } where g does not vanish:g ( z ) (cid:44) , and for every tangent vector ξ ∈ T z K N + (cid:27) K N + which does not lie in the line of z: ξ (cid:60) K · z , restricting on the subspace: A λ ( K N + ) ⊂ A ( K N + ) , the evaluation maps ( g · v ) z and d z ( g · )( ξ ) are K -linearly independent. In other words, the map: (cid:32) ( g · v ) z d z ( g · )( ξ ) (cid:33) : A λ ( K N + ) −→ K is surjective.Proof. This is a direct consequence of formula (129) and of the preceding lemma. (cid:3)
Codimensions of a ffi ne cones. Usually, it is more convenient to count dimension in an Eu-clidian space rather than in a projective space. Therefore we carry out the following lemma (cf.[37, p. 12, exercise 2.10]), which is geometrically obvious, as one point ( dim K =
0) in the projectivespace P N K corresponds to one K -line ( dim K =
1) in K N + . Lemma 8.12.
In an algebraically closed field K , let π : K N + → P N K be the canonical projection,and let: Y ⊂ P N K be a nonempty algebraic set defined by a homogeneous ideal:I ⊂ K [ z , . . . , z N ] . Denote by C ( Y ) the a ffi ne cone over Y:C ( Y ) : = π − ( Y ) ∪ { } ⊂ K N + . Then C ( Y ) is an algebraic set in K N + which is also defined by the ideal I (considered as anordinary ideal in K [ z , . . . , z N ] ), and it has dimension one more than Y: dim C ( Y ) = dim Y + . In other words, they have the same codimension: codim C ( Y ) = codim Y . (cid:3) The essence of the above geometric lemma is the following theorem in commutative algebra (cf.[42, p. 73, Cor. 5.21]): heorem 8.13. Let B be a homogeneous algebra over a field K , then: dim Spec B = dim Proj B + . (cid:3) Full rank of hypersurface equation matrices.
In an algebraically closed field K , for allpositive integers N (cid:62)
2, for all integers e = · · · N , for all positive integers (cid:15) , . . . , (cid:15) e (cid:62) d (cid:62)
1, consider the following e hypersurfaces: H , . . . , H e ⊂ P N K , each being defined as the zero set of a degree ( d + (cid:15) i ) Fermat-type homogeneous polynomial: F i = N (cid:88) j = A ji z dj ( i = ··· e ) , (130)where all A ji ∈ A (cid:15) i ( K N + ) are some degree (cid:15) i homogeneous polynomials.Now, denote by H the e × ( N +
1) matrix whose i -th row copies the ( N +
1) terms of F i in theexact order, i.e. the ( i , j )-th entries of H are: H i , j = A j − i z dj − i = ··· e ; j = ··· N + , so H writes as: H : = A z d · · · A N z dN ... . . . ... A e z d · · · A Ne z dN , (131)which we call the hypersurface equation matrix of F , . . . , F e . Passim , remark that by (130), thesum of all columns of H vanishes at every point [ z ] ∈ X : = H ∩ · · · ∩ H e .Also introduce: P ( M ) : = P (cid:16) ⊕ (cid:54) i (cid:54) e (cid:54) j (cid:54) N A (cid:15) i ( K N + ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : M (cid:17) the projectivized parameter space of ( A ji ) (cid:54) i (cid:54) e (cid:54) j (cid:54) N ∈ M .First, let us recall a classical theorem (cf. [59, p. 57, Theorem 2]) that somehow foreshadowsRemmert’s proper mapping theorem. Theorem 8.14.
The image of a projective variety under a regular map is closed. (cid:3)
The following lemma was proved by Brotbek in another version [7, p. 36, Proof of Claim 1],and the proof there is in a ffi ne coordinates ( z z j , . . . , (cid:98) z j z j , . . . , z N z j ): K N (cid:27) { z j (cid:44) } ⊂ P N K ( j = ··· N ) . Here, we may present a proof by much the same arguments in ambient coordinates ( z , . . . , z N ): K N + \ { } −→ P N K ( z , . . . , z N ) (cid:55)−→ [ z : · · · : z N ] . Lemma 8.15. In P ( M ) , there exists a proper algebraic subset: Σ (cid:36) P ( M ) uch that, for every choice of parameter outside Σ : (cid:104)(cid:0) A ji (cid:1) (cid:54) i (cid:54) e (cid:54) j (cid:54) N (cid:105) ∈ P ( M ) \ Σ , on the corresponding intersection: X = H ∩ · · · ∩ H e ⊂ P N K , the matrix H has full rank e everywhere: rank K H ( z ) = e ( ∀ [ z ] ∈ X ) . Sharing the same spirit as the famous
Fubini principle in combinatorics, the essence of the proofbelow is to count dimension in two ways, which is a standard method in algebraic geometry havingvarious forms (e.g. the proof of Bertini’s Theorem in [37, p. 179], main arguments in [18, 7], etc).
Proof.
Now, introduce the universal family X (cid:44) → P ( M ) × P N K of the intersections of such e Fermat-type hypersurfaces: X : = (cid:110)(cid:0) [ A ji ] , [ z ] (cid:1) ∈ P ( M ) × P N K : (cid:80) Nj = A ji z dj = , for i = · · · e (cid:111) , and then consider the subvariety B ⊂ X that consists of all ‘bad points’ defined by: rank K H (cid:54) e − . (132)Let π , π below be the two canonical projections: P ( M ) × P N K π (cid:121) (cid:121) π (cid:36) (cid:36) P ( M ) P N K . Since P ( M ) × P N K ⊃ B is a projective variety and π is a regular map, now applying Theorem 8.14,we see that: π ( B ) ⊂ P ( M )is an algebraic subvariety. Hence it is necessary and su ffi cient to show that: π ( B ) (cid:44) P ( M ) . (133)Our strategy is as follows. Step . To decompose P N K into a union of quasi-subvarieties: P N K = ∪ Nk = k P N K ◦ , (134)where k P N K ◦ consists of points [ z ] = [ z : z : · · · : z N ] ∈ P N K with exactly k vanishing homogeneouscoordinates, the other ones being nonzero. Step . For every integer k = · · · N , for every point [ z ] ∈ k P N K ◦ , to establish the fibre dimensionidentity: dim π − ([ z ]) ∩ B = dim P ( M ) − (cid:0) max { N − k − e + , } + e (cid:1) . (135) roof of Step . Without loss of generality, we may assume that the last k homogeneous coordi-nates of [ z ] vanish: z N − k + = · · · = z N = , (136)and then by the definition of k P N K ◦ , none of the first ( N − k +
1) coordinates z , . . . , z N − k vanish.Noting that: π − ([ z ]) ∩ B = π (cid:16) π − ([ z ]) ∩ B (cid:17)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) by Theorem 8.14 is an algebraic set × (cid:8) [ z ] (cid:9)(cid:124)(cid:123)(cid:122)(cid:125) one point set , and considering the canonical projection: (cid:98) π : M \ { } −→ P ( M ) , we receive: dim π − ([ z ]) ∩ B = dim π (cid:16) π − ([ z ]) ∩ B (cid:17) [use Lemma 8.12] = dim (cid:98) π − (cid:16) π (cid:0) π − ([ z ]) ∩ B (cid:1)(cid:17) ∪ { } − . (137)Now, observe that whatever choice of parameters: (cid:0) A ji (cid:1) (cid:54) i (cid:54) e (cid:54) j (cid:54) N ∈ M , the vanishing of the last k coordinates of [ z ] in (136) makes the last k columns of H ( z ) in (131)vanish. It is therefore natural to introduce the submatrix N + − k H of H that consists of the remainingnonvanishing columns, i.e. the first ( N + − k ) ones. Since the sum of all columns of H ( z ) vanishesby (130), the sum of all columns of N + − k H ( z ) also vanishes.Observe that the set: M ⊃ (cid:98) π − (cid:16) π (cid:0) π − ([ z ]) ∩ B (cid:1)(cid:17) ∪ { } = (cid:110)(cid:0) A ji (cid:1) (cid:54) i (cid:54) e (cid:54) j (cid:54) N ∈ M : sum of all the columns of N + − k H ( z ) vanishes,and rank K N + − k H ( z ) (cid:54) e − (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) from (132) (cid:111) is nothing but the inverse image of: Σ e , N + − ke − ⊂ Mat e × ( N + − k ) ( K ) [use notation of Lemma 8.9] under the K -linear map: N − k + H z : M −→ Mat e × ( N + − k ) ( K )( A ji ) i , j (cid:55)−→ N − k + H ( z ) , which is surjective by Lemma 8.11.Therefore we have the codimension identity: codim (cid:98) π − (cid:16) π (cid:0) π − ([ z ]) ∩ B (cid:1)(cid:17) ∪ { } = codim Σ e , N + − ke − [ N − k + H z is linear and surjective][use Lemma (8.9) ] = max { N − k − e + , } + e , (138) nd thereby we receive: dim π − ([ z ]) ∩ B = dim (cid:98) π − (cid:16) π (cid:0) π − ([ z ]) ∩ B (cid:1)(cid:17) ∪ { } − [use (137) ][by definition of codimension] = dim M − codim (cid:98) π − (cid:16) π (cid:0) π − ([ z ]) ∩ B (cid:1)(cid:17) ∪ { } − [exercise] = dim P ( M ) − codim (cid:98) π − (cid:16) π (cid:0) π − ([ z ]) ∩ B (cid:1)(cid:17) ∪ { } [use (138) ] = dim P ( M ) − (cid:0) max { N − k − e + , } + e (cid:1) , which is exactly our claimed fibre dimension identity (135). (cid:3) Step . Applying Lemma 8.2 to the regular map: π : π − (cid:0) k P N K ◦ (cid:1) ∩ B −→ k P N K ◦ , remembering: dim k P N K ◦ = N − k ( k = ··· N ) , together with the identity (135), we receive the dimension estimate: dim π − (cid:0) k P N K ◦ (cid:1) ∩ B (cid:54) dim k P N K ◦ + dim P ( M ) − max { N − k − e + , } − e (cid:54) ( N − k ) + dim P ( M ) − ( N − k − e + − e = dim P ( M ) − . (139)Note that B can be written as the union of ( N +
1) quasi-subvarieties: B = π − (cid:0) P N K (cid:1) ∩ B = π − (cid:0) ∪ Nk = k P N K ◦ (cid:1) ∩ B = (cid:16) ∪ Nk = π − (cid:0) k P N K ◦ (cid:1)(cid:17) ∩ B = ∪ Nk = (cid:16) π − (cid:0) k P N K ◦ (cid:1) ∩ B (cid:17) , each one being, thanks to (139), of dimension less than or equal to: dim P ( M ) − , and therefore we have the dimension estimate: dim B (cid:54) dim P ( M ) − . Finally, (133) follows from the dimensional comparison: dim π ( B ) (cid:54) dim B (cid:54) dim P ( M ) − . (cid:3) In the more general context of our moving coe ffi cients method, we now want to have an every-where full-rank property analogous to Lemma 8.15 just obtained.Observing that in (92), the number of terms in each polymonial F i is:( N + + N (cid:88) (cid:96) = c + r + (cid:32) N + (cid:96) + (cid:33) ( (cid:96) + , nd recalling that the K -linear subspace A (cid:15) i ( K N + ) ⊂ K [ z , . . . , z N ] spanned by all degree (cid:15) i homo-geneous polynomials is of dimension: dim K A (cid:15) i ( K N + ) = (cid:32) N + (cid:15) i N (cid:33) , we may denote by P (cid:169) K the projectivized parameter space of such c + r hypersurfaces, with theinteger: (cid:169) : = (cid:20) ( N + + N (cid:88) (cid:96) = c + r + (cid:32) N + (cid:96) + (cid:33) ( (cid:96) + (cid:21) c + r (cid:88) i = (cid:32) N + (cid:15) i N (cid:33) . (140)Now, by mimicking the construction of the matrix H in (131), employing the notation in Sub-section 7.2, for every integer ν = · · · N , let us denote by H ν the ( c + r ) × ( N +
1) matrix whose i -th row copies the ( N +
1) terms of F i in (105). Also, for every integer τ = · · · N −
1, for everyindex ρ = τ + · · · N , let us denote by H τ,ρ the ( c + r ) × ( N +
1) matrix whose i -th row copies the( N +
1) terms of F i in (107). Lemma 8.16. In P (cid:169) K , there exists a proper algebraic subset: Σ (cid:36) P (cid:169) K such that, for every choice of parameter outside Σ : (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ Σ , on the corresponding intersection: X = H ∩ · · · ∩ H c + r ⊂ P N K , all the matrices H ν , H τ,ρ have full rank c: rank K H ν ( z ) = c + r , rank K H τ,ρ ( z ) = c + r ( ∀ [ z ] ∈ X ) . We can copy the proof of Lemma 8.15 without much modification and thus everything workssmoothly. Alternatively, we may present a short proof by applying Lemma 8.15.
Proof. Observation 1.
We need only prove this lemma separately for each matrix H ν (resp. H τ,ρ ),i.e. to show that there exists a proper algebraic subset: Σ ν (resp. Σ τ,ρ ) (cid:36) P (cid:169) K outside of which every choice of parameter succeeds. Then the union of all these proper algebraicsubsets works: Σ : = ∪ N ν = Σ ν ∪ ∪ τ = ··· N − ρ = τ + ··· N Σ τ,ρ (cid:36) P (cid:169) K . Observation 2.
For each matrix H ν (resp. H τ,ρ ), inspired by the beginning arguments in the proofof Lemma 8.15, especially (133), we only need to find one parameter: (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ Σ with the desired property. Observation 3.
Now, setting all the moving coe ffi cients zero: M •• : = , hanks to (103), (106), the equations (105) become exactly the equations (130), and therefore allthe matrices H ν become the same matrix H of Lemma 8.15 (with e = c + r ). Similarly, so do all thematrices H τ,ρ . Observation 4.
Now, a direct application of Lemma 8.15 clearly yields more than one parameter,an infinity! (cid:3)
Once again, by mimicking the construction of the matrix H in Lemma 8.15, employing thenotation in subsection 7.3, let us denote by v ,..., v η H ν (resp. v ,..., v η H τ,ρ ) the c × ( N +
1) matrix whose i -th row copies the ( N +
1) terms of F i in (120) (resp. (121)). Lemma 8.17. In P (cid:169) K , there exists a proper algebraic subset: v ,..., v η Σ (cid:36) P (cid:169) K such that, for every choice of parameter outside v ,..., v η Σ : (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ v ,..., v η Σ on the corresponding intersection: X = H ∩ · · · ∩ H c + r ⊂ P N K , all the matrices v ,..., v η H ν and v ,..., v η H τ,ρ have full rank c + r: rank K v ,..., v η H ν ( z ) = c + r , rank K v ,..., v η H τ,ρ ( z ) = c + r ( ∀ [ z ] ∈ X ) . (cid:3) The proof goes exactly the same way as in the preceding lemma.9.
Controlling the base locus
Characterization of the base locus.
Now, we are in a position to characterize the base locusof all the obtained global twisted symmetric di ff erential n -forms in (110), (111): BS : = Base Locus of { φ ν j ,..., j n , ψ τ,ρ j ,..., j n } ν,τ,ρ (cid:54) j < ··· < j n (cid:54) c ⊂ P (cid:0) T V (cid:12)(cid:12)(cid:12) X (cid:1) , (141)where P (cid:0) T V (cid:12)(cid:12)(cid:12) X (cid:1) ⊂ P (T P N K ) is given by: P (cid:0) T V (cid:12)(cid:12)(cid:12) X (cid:1) : = (cid:110) ([ z ] , [ ξ ]) : F i ( z ) = , dF j (cid:12)(cid:12)(cid:12) z ( ξ ) = , ∀ i = · · · c + r , ∀ j = · · · c (cid:111) To begin with, for every ν = · · · N , let us study the specific base locus: BS ν : = Base Locus of { φ ν j ,..., j n } (cid:54) j < ··· < j n (cid:54) c ⊂ P (cid:0) T V (cid:12)(cid:12)(cid:12) X (cid:1) associated with only the twisted symmetric di ff erential forms obtained in (110).For each sequence of ascending indices:1 (cid:54) j < · · · < j n (cid:54) c , by mimicking the construction of the matrices K , (cid:98) K j ,..., j n ; j at the end of Subsection 6.3, in accor-dance with the first kind of manipulation (105), we construct the ( c + r + c ) × ( N +
1) matrix K ν inthe obvious way, i.e. by copying terms, di ff erentials, and then we define the analogous (cid:98) K ν j ,..., j n ; j .First, let us look at points (cid:0) [ z ] , [ ξ ] (cid:1) ∈ BS ν having all coordinates nonvanishing: z · · · z N (cid:44) . (142) or each symmetric horizontal di ff erential n -form (cid:98) φ ν j ,..., j n which corresponds to φ ν j ,..., j n in the senseof Propositions 6.10, 6.9, for every j = · · · N , we receive:0 = (cid:98) φ ν j ,..., j n ; j ( z , ξ ) [since ([ z ] , [ ξ ]) ∈ BS ν ][use (79) ] = ( − j z (cid:63) · · · z (cid:63) N (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:44) det (cid:0)(cid:98) K ν j ,..., j n ; j (cid:1) ( z , ξ ) , where all integers (cid:63) are of no importance here. Indeed, we can drop the nonzero factor ( − j z (cid:63) ··· z (cid:63) N andobtain: det (cid:0) (cid:98) K ν j ,..., j n ; j (cid:124) (cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32) (cid:125) N × N matrix (cid:1) ( z , ξ ) = . In other words: rank K (cid:98) K ν j ,..., j n ; j ( z , ξ ) (cid:54) N − . Now, letting the index j run from 0 to N , we receive: rank K K ν j ,..., j n ( z , ξ ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) N × ( N +
1) matrix (cid:54) N − , (143)where K ν j ,..., j n is defined analogously to the matrix C ν j ,..., j n before Proposition 6.8 in the obviousway.Note that the first c + r rows of K ν j ,..., j n constitute the matrix H ν in Lemma 8.16, which assertsthat for a generic choice of parameter: rank K H ν ( z ) = c + r . Now, in (143), letting 1 (cid:54) j < · · · < j n (cid:54) c vary, and applying Lemma 8.5, we immediatelyreceive: rank K K ν ( z , ξ ) (cid:54) N − . Conversely, it is direct to see that any point (cid:0) [ z ] , [ ξ ] (cid:1) ∈ X P (T V ) satisfying this rank inequality liesin the base locus BS ν .Note that a point (cid:0) [ z ] , [ ξ ] (cid:1) ∈ P (T P N ) lies in P (cid:0) T V (cid:12)(cid:12)(cid:12) X (cid:1) if and only if the sum of all columns of K ν ( z , ξ ) vanishes. Summarizing the above analysis, restricting to the coordinates nonvanishingpart of P (T P N K ): P ◦ (T P N K ) : = P (T P N K ) ∩ { z · · · z N (cid:44) } , we conclude the following generic characterization of: BS ν ∩ P ◦ (T P N K ) , where the exceptional locus Σ just below is defined in Lemma 8.16. Proposition 9.1.
For every choice of parameter outside Σ : (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ Σ a point: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ P ◦ (T P N K ) lies in the base locus: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ BS ν f and only if: rank K K ν ( z , ξ ) (cid:54) N − , and the sum of all columns vanishes . (cid:3) Now, for every integer τ = · · · N − ρ = τ + · · · N , the base locus: BS τ,ρ : = Base Locus of { ψ τ,ρ j ,..., j n } (cid:54) j < ··· < j n (cid:54) c ⊂ P (cid:0) T V (cid:12)(cid:12)(cid:12) X (cid:1) associated with the twisted symmetric di ff erential forms obtained in (111) enjoys the followinggeneric characterization on the coordinates nonvanishing set { z · · · z N (cid:44) } . Of course, the matrix K τ,ρ is defined analogously to the matrix K ν in the obvious way. A repetition of the precedingarguments yields: Proposition 9.2.
For every choice of parameter outside Σ : (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ Σ a point: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ P ◦ (T P N K ) lies in the base locus: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ BS τ,ρ if and only if: rank K K τ,ρ ( z , ξ ) (cid:54) N − , and the sum of all columns vanishes . (cid:3) It is now time to clarify the (uniform) structures of the matrices K ν , K τ,ρ .Thanks to the above two Propositions 9.1, 9.2, we may now receive a generic characterizationof: BS ∩ P ◦ (T P N ) . Firstly, we construct the ( c + r + c ) × (2 N +
2) matrix M such that, for i = · · · c + r , j = · · · c ,its i -row copies the (2 N +
2) terms of F i in (102) in the exact order, and its ( c + r + j )-th row is thedi ff erential of the j -th row. In order to distinguish the first ( N +
1) ‘dominant’ columns from thelast ( N +
1) columns of moving coe ffi cient terms, we write M as: M = (cid:16) A | · · · | A N | B | · · · | B N (cid:17) . For every index ν = · · · N , comparing (105), (106) with (102), the matrix K ν is nothing but: K ν = (cid:16) A | · · · | (cid:98) A ν | · · · | A N | A ν + N (cid:88) j = B j (cid:17) . (144)Similarly, for every integer τ = · · · N − ρ = τ + · · · N , comparing (107),(108) with (102), the matrix K τ,ρ is nothing but: K τ,ρ = (cid:16) A + B | · · · | A τ + B τ | A τ + | · · · | (cid:98) A ρ | · · · | A N | A ρ + N (cid:88) j = τ + B j (cid:17) . (145)Secondly, we introduce the algebraic subvariety: M N c + r ⊂ Mat (2 c + r ) × N + ( K ) (146)consisting of all ( c + r + c ) × N +
1) matrices ( α | α | · · · | α N | β | β | · · · | β N ) such that: i) the sum of these (2 N +
2) colums is zero: α + α + · · · + α N + β + β + · · · + β N = ; (147) (ii) for every index ν = · · · N , replacing α ν with α ν + ( β + β + · · · + β N ) in the collection ofcolumn vectors { α , α , . . . , α N } , there holds the rank inequality: rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α N , α ν + ( β + β + · · · + β N ) (cid:9) (cid:54) N −
1; (148) (iii) for every integer τ = · · · N −
1, for every index ρ = τ + · · · N , replacing α ρ with α ρ + ( β τ + + · · · + β N ) in the collection of column vectors { α + β , . . . , α τ + β τ , α τ + , . . . , α ρ , . . . , α N } , thereholds the rank inequality: rank K (cid:8) α + β , α + β , . . . , α τ + β τ , α τ + , . . . , (cid:98) α ρ , . . . , α N , α ρ + ( β τ + + · · · + β N ) (cid:9) (cid:54) N − . (149) Proposition 9.3.
For every choice of parameter outside Σ : (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ Σ a point: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ P ◦ (T P N ) lies in the base locus: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ BS if and only if: M ( z , ξ ) ∈ M N c + r . (cid:3) Furthermore, for all integers 1 (cid:54) η (cid:54) n −
1, for every sequence of ascending indices:0 (cid:54) v < · · · < v η (cid:54) N , we also have to analyze the base locus of the twisted symmetric di ff erential forms (122), (123): v ,..., v η BS : = Base Locus of { v ,..., v η φ ν j ,..., j n − η , v ,..., v η ψ τ,ρ j ,..., j n − η } ν,τ,ρ (cid:54) j < ··· < j n − η (cid:54) c (150)in the intersection of the η hyperplanes: v ,..., v η P (T P N ) : = P (T P N ) ∩ { z v = · · · = z v η = } , and more specifically, we focus on the ‘interior part’: v ,..., v η P ◦ (T P N ) : = v ,..., v η P (T P N ) ∩ { z r · · · z r N − η (cid:44) } [see (80) for the indices r , . . . , r N − η ] . Firstly, we construct the ( c + r + c ) × (2 N + − η ) matrix v ,..., v η M , which will play the same roleas the matrix M , whose i -row ( i = · · · c + r ) copies the (2 N + − η ) terms of (114) in the exactorder, and whose ( c + r + j )-th row ( j = · · · c ) is the di ff erential of the j -th row.Secondly, in correspondence with M N c + r , by replacing plainly N with N − η , we introduce thealgebraic variety: M N − η c + r ⊂ Mat (2 c + r ) × N − η + ( K ) . (151)Thirdly, let us recall the exceptional subvatiety: v ,..., v η Σ (cid:36) P (cid:169) K defined in Proposition 8.17.By performing the same reasoning as in the preceding proposition, we get: roposition 9.4. For every choice of parameter outside v ,..., v η Σ : (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ v ,..., v η Σ a point: (cid:0) [ z ] , [ ξ ] (cid:1) ∈ v ,..., v η P ◦ (T P N ) lies in the base locus (150) : (cid:0) [ z ] , [ ξ ] (cid:1) ∈ v ,..., v η BS if and only if: v ,..., v η M ( z , ξ ) ∈ M N − η c + r . (cid:3) Emptiness of the base loci.
First, for the algebraic varieties (146), (151), we claim the fol-lowing codimension estimates, which serve as the engine of the moving coe ffi cients method. How-ever, we will not present it here but in the next section. Lemma 9.5 ( Core Lemma of MCM). (i)
For every positive integers N (cid:62) , for every integersc , r (cid:62) with c + r (cid:62) N, there holds the codimension estimate: codim M N c + r (cid:62) dim P ◦ (T P N ) = N − . (ii) For every positive integer η = · · · N − ( c + r ) − , for every sequence of ascending indices: (cid:54) v < · · · < v η (cid:54) N , there holds the codimension estimate: codim M N − η c + r (cid:62) dim v ,..., v η P ◦ (T P N ) = N − η − . (cid:3) Now, let us show the power of this Core Lemma.Bearing Proposition 9.3 in mind, by mimicking the proof of Proposition 8.15, it is natural tointroduce the subvariety: M N c + r (cid:44) → P (cid:169) K × P ◦ (T P N ) , which is defined ‘in family’ by: M N c + r : = (cid:110)(cid:0) [ A •• , M •• ]; [ z ] , [ ξ ] (cid:1) ∈ P (cid:169) K × P ◦ (T P N ) : M ( z , ξ ) ∈ M N c + r (cid:111) . Proposition 9.6.
There holds the dimension estimate: dim M N c + r (cid:54) dim P (cid:169) K . Proof.
Let π , π be the two canonical projections: P (cid:169) K × P ◦ (T P N ) π (cid:122) (cid:122) π (cid:37) (cid:37) P (cid:169) K P ◦ (T P N ) . By mimicking Step 2 in Lemma 8.15, for every point ([ z ] , [ ξ ]) ∈ P ◦ (T P N ), we claim the fibre dimen-sion estimate: dim π − ([ z ] , [ ξ ]) ∩ M N c + r = dim P (cid:169) K − codim M N c + r (152) roof. Noting that: π − ([ z ] , [ ξ ]) ∩ M N c + r = π (cid:16) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:17)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) by Theorem 8.14 is an algebraic set × (cid:8) ([ z ] , [ ξ ]) (cid:9)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) one point set , and considering the canonical projection: (cid:98) π : K (cid:169) \ { } −→ P (cid:169) K , we receive: dim π − ([ z ] , [ ξ ]) ∩ M N c + r = dim π (cid:16) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:17) [use Lemma 8.12] = dim (cid:98) π − (cid:16) π (cid:0) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:1)(cid:17) ∪ { } − . (153)Now, notice that the set: K (cid:169) ⊃ (cid:98) π − (cid:16) π (cid:0) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:1)(cid:17) ∪ { } = (cid:110)(cid:0) A •• , M •• (cid:1) ∈ K (cid:169) : M ( z , ξ ) ∈ M N c + r (cid:111) is nothing but the inverse image of: M N c + r ⊂ Mat (2 c + r ) × N + ( K )under the K -linear map: M z , ξ : K (cid:169) −→ Mat (2 c + r ) × N + ( K ) (cid:0) A •• , M •• (cid:1) (cid:55)−→ M ( z , ξ ) , which is surjective by the construction of M — see (102), (103), and by applying Lemma 8.11 —since z (cid:44) , . . . , z N (cid:44) ξ (cid:60) K · z .Therefore, we have the codimension identity: codim (cid:98) π − (cid:16) π (cid:0) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:1)(cid:17) ∪ { } = codim M N c + r [M z ,ξ is linear and surjective] , (154)and thereby we receive: dim π − ([ z ] , [ ξ ]) ∩ M N c + r = dim (cid:98) π − (cid:16) π (cid:0) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:1)(cid:17) ∪ { } − [use (153) ][by definition of codimension] = dim K (cid:169) − codim (cid:98) π − (cid:16) π (cid:0) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:1)(cid:17) ∪ { } − [why?] = dim P (cid:169) K − codim (cid:98) π − (cid:16) π (cid:0) π − ([ z ] , [ ξ ]) ∩ M N c + r (cid:1)(cid:17) ∪ { } [use (154) ] = dim P (cid:169) K − codim M N c + r , which is exactly our claimed fibre dimension identity. (cid:3) Lastly, by applying the Fibre Dimension Estimate 8.2, we receive: dim M N c + r (cid:54) dim P ◦ (T P N ) + dim P (cid:169) K − codim M N c + r [use (152) ][use Core Lemma 9.5] (cid:54) dim P ◦ (T P N ) + dim P (cid:169) K − dim P ◦ (T P N ) = dim P (cid:169) K , which is our claimed dimension estimate. (cid:3) ow, restricting the canonical projection π to M N c + r : π : M N c + r −→ P (cid:169) K , according to the dimension inequality just obtained, we gain: Proposition 9.7.
There exists a proper algebraic subset Σ (cid:48) (cid:36) P (cid:169) K such that, for every choice ofparameter outside Σ (cid:48) : P = (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ Σ (cid:48) , the intersection of the fibre π − ( P ) with M N c + r is discrete or empty: dim π − ( P ) ∩ M N c + r (cid:54) . (cid:3) Combining Propositions 9.3 and 9.7, we receive:
Proposition 9.8.
Outside the proper algebraic subset: Σ ∪ Σ (cid:48) (cid:36) P (cid:169) K , for every choice of parameter: (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ ( Σ ∪ Σ (cid:48) ) , the base locus in the coordinates nonvanishing set: BS ∩ { z · · · z N (cid:44) } is discrete or empty. (cid:3) Moreover, bearing in mind Proposition 9.4, by repeating the same reasoning as in the precedingproposition, consider the subvariety: v ,..., v η M N − η c + r (cid:44) → P (cid:169) K × v ,..., v η P ◦ (T P N )which is defined ‘in family’ by: v ,..., v η M N c + r : = (cid:110)(cid:0) [ A •• , B •• ]; [ z ] , [ ξ ] (cid:1) ∈ P (cid:169) K × v ,..., v η P ◦ (T P N ) : v ,..., v η M ( z , ξ ) ∈ M N − η c + r (cid:111) , and hence receive a very analog of Proposition 9.6. Proposition 9.9.
There holds the dimension estimate: dim v ,..., v η M N − η c + r (cid:54) dim P (cid:169) K . (cid:3) Again, restricting the canonical projection π to v ,..., v η M N − η c + r : π : v ,..., v η M N − η c + r −→ P (cid:169) K , according to the dimension inequality above, we receive: Proposition 9.10.
There exists a proper algebraic subset v ,..., v η Σ (cid:48) (cid:36) P (cid:169) K such that, for every choiceof parameter outside v ,..., v η Σ (cid:48) : P = (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ v ,..., v η Σ (cid:48) , the intersection of the fibre π − ( P ) with M N − η c + r is discrete or empty: dim π − ( P ) ∩ M N − η c + r (cid:54) . (cid:3) Combining now Propositions 9.4 and 9.10, we receive: roposition 9.11. Outside the proper algebraic subset: v ,..., v η Σ ∪ v ,..., v η Σ (cid:48) (cid:36) P (cid:169) K for every choice of parameter: (cid:2) A •• , M •• (cid:3) ∈ P (cid:169) K \ ( v ,..., v η Σ ∪ v ,..., v η Σ (cid:48) ) , the base locus in the corresponding ‘coordinates nonvanishing’ set: v ,..., v η BS ∩ { z r · · · z r N − η (cid:44) } is discrete or empty. (cid:3) The Engine of MCM
Core Codimension Formulas.
Our motivation of this section is to prove the Core Lemma 9.5,which will succeed in Subsection 10.6.As an essential step, by induction on positive integers p (cid:62) (cid:54) (cid:96) (cid:54) p , we first estimate thecodimension (cid:96) C p of the algebraic variety: (cid:96) X p ⊂ Mat p × p ( K ) (155)which consists of p × p matrices X p = ( α , . . . , α p , β , . . . , β p ) such that: (i) the first p column vectors have rank: rank K (cid:8) α , . . . , α p (cid:9) (cid:54) (cid:96) ; (156) (ii) for every index ν = · · · p , replacing α ν with α ν + ( β + · · · + β p ) in the collection of columnvectors { α , . . . , α p } , there holds the rank inequality: rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α p , α ν + ( β + · · · + β p ) (cid:9) (cid:54) p −
1; (157) (iii) for every integer τ = · · · p −
1, for every index ρ = τ + · · · p , replacing α ρ with α ρ + ( β τ + + · · · + β p ) in the collection of column vectors { α + β , . . . , α τ + β τ , α τ + , . . . , α ρ , . . . , α p } , thereholds the rank inequality: rank K (cid:8) α + β , . . . , α τ + β τ , α τ + , . . . , (cid:98) α ρ , . . . , α p , α ρ + ( β τ + + · · · + β p ) (cid:9) (cid:54) p − . (158)Let us start with the easy case (cid:96) = Proposition 10.1.
For every integer p (cid:62) , the codimension value (cid:96) C p for (cid:96) = is: C p = p + . (159) Proof.
Now, (i) is equivalent to: α = · · · = α p = (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = p . Thus (ii) holds trivially, and the only nontrivial inequality in (iii) is: rank K (cid:8) + β , . . . , + β p (cid:9) (cid:54) p − (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = . , which contributes one more codimension. (cid:3) For the general case (cid:96) = · · · p , we will use Gaussian eliminations and do inductions on p , (cid:96) .First, let us count the codimension of the exceptional locus of Gaussian eliminations. roposition 10.2. For every integer p (cid:62) , the codimensions (cid:96) C p of the algebraic varieties: { α + β = } ∩ (cid:96) X p ⊂ Mat p × p ( K ) read according to the values of (cid:96) as: (cid:96) C p = p + ( (cid:96) = p − , p ) , p + ( p − (cid:96) ) (cid:96) = ··· p − . The following lemma is the key ingredient for the proof.
Lemma 10.3.
In a field K , let W be a K -vector space. Let p (cid:62) be a positive integer. For any ( p + vectors: α , . . . , α p , β ∈ W , the rank restriction: rank K { α , . . . , (cid:98) α ν , . . . , α p , α ν + β } (cid:54) p − ( ν = ··· p ) , (160) is equivalent to either: rank K { α , . . . , α p , β } (cid:54) p − , or: rank K { α , . . . , α p } = p , ( α + · · · + α p ) + β = . Proof.
Since ‘ ⇐ = ’ is clear, we only prove the direction ‘ = ⇒ ’.We divide the proof according to the rank of { α , . . . , α p } into two cases. Case 1: rank K { α , . . . , α p } (cid:54) p −
1. Assume on the contrary that: rank K { α , . . . , α p , β } (cid:62) p . (161)Since we have the elementary estimate: rank K { α , . . . , α p , β } (cid:54) rank K { α , . . . , α p } + rank K { β } (cid:54) ( p − + = p , (162)the inequalities ‘ (cid:62) ’ or ‘ (cid:54) ’ in (161) and (162) are exactly equalities ‘ = ’, and thus we have: β (cid:60) Span K { α , . . . , α p } , (163) rank K { α , . . . , α p } = p − . Consequently, it is clear that we can find a certain index ν ∈ { , . . . , p } such that: rank K { α , . . . , (cid:98) α ν , . . . , α p } = p − , whence the above rank inequality (160) implies: α ν + β ∈ Span K { α , . . . , (cid:98) α ν , . . . , α p } , (164)which contradicts the formula (163). Case 2: rank K { α , . . . , α p } = p . Here, inequalities (160) also yield (164) for every ν , whence: β + ( α + · · · + α p ) ∈ Span K { α , . . . , (cid:98) α ν , . . . , α p } . Now, letting ν run from 1 to p , and noting that: ∩ p ν = Span K { α , . . . , (cid:98) α ν , . . . , α p } = { } , we immediately conclude the proof. (cid:3) roof of Proposition 10.2. For every matrix X p = ( α , . . . , α p , β , . . . , β p ) such that: α + β = (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = p , (165)the conditions (158) in (iii) is trivial, and the restriction (157), thanks to the lemma just obtained,is equivalent either to: rank K { α , . . . , α p , β + · · · + β p } (cid:54) p − , (166)or to: rank K { α , . . . , α p } = p , β + · · · + β p = − ( α + · · · + α p ) . (167)Now, since α + β =
0, adding the first column vector of (166) to the last one, we get: rank K { α , . . . , α p , β + · · · + β p } (cid:54) p − (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = . , and similarly, (167) is equivalent to: rank K { α , . . . , α p } = p , ( α + · · · + α p ) + ( β + · · · + β p ) = (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = p . Therefore, when (cid:96) = p − (cid:96) = p , we obtain the codimension formulas: p − C p = p + , p C p = min { p + , p + p } = p + . When (cid:96) = · · · p −
2, the restriction (ii) is a consequence of (i) : rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α p , α ν + ( β + · · · + β p ) (cid:9) (cid:54) rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α p (cid:9) + rank K (cid:8) α ν + ( β + · · · + β p ) (cid:9) (cid:54) rank K (cid:8) α , . . . , α p (cid:9) + (cid:54) (cid:96) + (cid:54) p − . Lastly, applying Lemma 8.8, restriction (i) contributes codimension ( p − (cid:96) ) . Together with (165),this finishes the proof. (cid:3) Now, we claim the following
Codimension Induction Formulas , the proof of which will appearin Subsection 10.5. In order to make sense of (cid:96) − C p − in (170) when (cid:96) =
1, we henceforth make a convention: − C p − : = ∞ . Proposition 10.4 ( Codimension Induction Formulas). (i)
For every positive integer p (cid:62) , for (cid:96) = p, the codimension value p C p satisfies: p C p = min (cid:8) p , p − C p (cid:9) . (168) (ii) For every positive integer p (cid:62) , for (cid:96) = p − , the codimension value (cid:96) C p satisfies: p − C p (cid:62) min (cid:8) p − C p , p − C p − + , p − C p − + , p − C p − (cid:9) . (169) (iii) For all integers (cid:96) = · · · p − , the codimension values (cid:96) C p satisfy: (cid:96) C p (cid:62) min (cid:8) (cid:96) C p , (cid:96) C p − + p − (cid:96) ) − , (cid:96) − C p − + ( p − (cid:96) ) , (cid:96) − C p − (cid:9) . (170) n fact, all the above inequalities ‘ (cid:62) ’ should be exactly equalities ‘ = ’. Nevertheless, ‘ (cid:62) ’ arealready adequate for our purpose.Now, let us establish the initial data for the induction process. Proposition 10.5.
For the initial case p = , there hold the codimension values: C = , C = , C = . Proof.
Recalling formulas (159) and (168), we only need to prove C = α , α , β , β ) ∈ X \ X , we have: rank K (cid:8) α , α (cid:9) = , (171) rank K (cid:8) α + ( β + β ) , α (cid:9) (cid:54) , (172) rank K (cid:8) α , α + ( β + β ) (cid:9) (cid:54) , (173) rank K (cid:8) α + β , α + β (cid:9) (cid:54) . (174)Either α or α is nonzero. Firstly, assume α (cid:44) . Then (171) yields: α ∈ K · α , (175)and (173) yields: α + ( β + β ) ∈ K · α , whence by subtracting we receive: β + β ∈ K · α . (176)Next, adding the second column vector of (174) to the first one, we see: rank K (cid:8) α + α + ( β + β ) , α + β (cid:9) (cid:54) . (177)By (175) and (176): α + α + ( β + β ) ∈ K · α , therefore (177) yields two possible situations, the first one is: α + α + β + β = , (178)and the second one is α + α + β + β (cid:44) plus: α + β ∈ K · α . Recalling (175), the latter case immediately yields: β ∈ K · α , and then (176) implies: β ∈ K · α , thus: rank K (cid:8) α , α , β , β (cid:9) = . (179)Summarizing, the set: (cid:0) X \ X (cid:1) ∩ { α (cid:44) } s contained in the union of two algebraic varieties, the first one is defined by (175), (176), (178),and the second one is defined by (179). Since both of the two varieties are of codimension 3, weget: codim (cid:0) X \ X (cid:1) ∩ { α (cid:44) } (cid:62) . Secondly, by symmetry, we also have: codim (cid:0) X \ X (cid:1) ∩ { α (cid:44) } (cid:62) . Hence the union of the above two sets satisfies: codim X \ X (cid:62) . Now, recalling (159): codim X = > , we immediately receive: codim X (cid:62) . Finally, noting that X contains the subvariety: (cid:110) rank { α , α , β , β } (cid:54) (cid:111)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = . ⊂ Mat × ( K ) , it follows: codim X (cid:54) . In conclusion, the above two estimates squeeze out the desired codimension identity. (cid:3)
Admitting temporally Proposition 10.4, it is now time to deduce the crucial
Proposition 10.6 ( Core Codimension Formulas).
For all integers p (cid:62) , there hold the codimen-sion estimates: (cid:96) C p (cid:62) (cid:96) + ( p − (cid:96) ) + ( (cid:96) = ··· p − , (180) and the codimension identity: p C p = p . (180 (cid:48) ) Proof.
The case p = (cid:48) ) hold for some integer p − (cid:62) p .Firstly, formula (159) yields the case (cid:96) = (cid:96) = p −
1, thanks to Proposition 10.2 and to the induction hypothesis,formula (169) immediately yields: p − C p (cid:62) min (cid:8) p − C p , p − C p − + , p − C p − + , p − C p − (cid:9) (cid:62) min (cid:8) p + , ( p − + , ( p − + + + , ( p − + + (cid:9) = p + = ( p − + + . (181)Similarly, for (cid:96) = · · · p −
2, recalling formula (170): (cid:96) C p (cid:62) min (cid:8) (cid:96) C p , (cid:96) C p − + p − (cid:96) ) − , (cid:96) − C p − + ( p − (cid:96) ) , (cid:96) − C p − (cid:9) , nd computing: (cid:96) C p = p + ( p − (cid:96) ) = (cid:96) + ( p − (cid:96) ) + ( p − (cid:96) ) (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) (cid:62) , (cid:96) C p − + p − (cid:96) ) − (cid:62) (cid:2) (cid:96) + ( p − − (cid:96) ) + (cid:3) + p − (cid:96) ) − = (cid:96) + ( p − (cid:96) ) − p − (cid:96) ) + + + p − (cid:96) ) − = (cid:96) + ( p − (cid:96) ) + (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) the desired lower bound! , (cid:96) − C p − + ( p − (cid:96) ) (cid:62) (cid:2) ( (cid:96) − + ( p − (cid:96) ) + (cid:3) + ( p − (cid:96) ) = (cid:96) + ( p − (cid:96) ) + ( p − (cid:96) ) (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) (cid:62) , (cid:96) − C p − (cid:62) ( (cid:96) − + ( p − (cid:96) + + = ( (cid:96) − + (cid:2) ( p − (cid:96) ) + p − (cid:96) ) + (cid:3) + = (cid:96) + ( p − (cid:96) ) + p − (cid:96) ) (cid:124) (cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32) (cid:125) (cid:62) , we distinguish the desired lower bound without di ffi culty.Lastly, the formula (168) and (181) immediately yield (180 (cid:48) ): p C p = p , which concludes the proof. (cid:3) Remark 10.7.
In fact, the above estimates “ (cid:62) ” in (180) are exactly identities “ = ”. By the samereasoning, in Section 12, we will generalize the Core Codimension Formulas to cases of lessnumber of moving coe ffi cients terms, and thus receive better lower bounds on the hypersurfacesdegrees.10.2. Gaussian eliminations.
Following the notation in (155), we denote by: X p = ( α , . . . , α p , β , . . . , β p )the coordinate columns of Mat p × p ( K ), where each of the first p columns explicitly writes as: α i = ( α , i , . . . , α p , i ) T , and where each of the last p columns explicitly writes as: β i = ( β , i , . . . , β p , i ) T . First, observing the structures of the matrices in (157), (158): X ,ν p : = (cid:0) α | · · · | (cid:98) α ν | · · · | α p | α ν + ( β + · · · + β p ) (cid:1) , X τ,ρ p : = (cid:0) α + β | · · · | α τ + β τ | α τ + | · · · | (cid:98) α ρ | · · · | α p | α ρ + ( β τ + + · · · + β p ) (cid:1) , where, slightly di ff erently, the second underlined columns are understood to appear in the firstunderlined removed places, we realize that they have the uniform shapes: X ,ν p = X p I ,ν p , X τ,ρ p = X p I τ,ρ p , (182) here the 2 p × p matrices I ,ν p explicitly read as: p (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123) p p ν -th column the upper p × p submatrix being the identity, the lower p × p submatrix being zero except its ν -thcolumn being a column of 1, and where lastly, the 2 p × p matrices I τ,ρ p explicitly read as:111 1 1 11 1 p (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123) p τ p − τρ -th column τ -th column the upper p × p submatrix being the identity, the lower p × p submatrix being zero except τ copiesof 1 in the beginning diagonal and p − τ copies of 1 at the end of the ρ -th column. Observation 10.8.
For all p (cid:62) , τ = · · · p − , ρ = τ + · · · p, the matrices I τ,ρ p transform to I τ − ,ρ − p − after deleting the first column and the rows , p + . (cid:3) Next, observe that all matrices X τ,ρ have the same first column: α + β = ( α , + β , | · · · | α p , + β p , ) T . herefore, when α , + β , (cid:44)
0, operating Gaussian eliminations by means of the matrix: G : = − α , + β , α , + β , ... . . . − α p , + β p , α , + β , , (183)these matrices X τ,ρ become simpler: G X τ,ρ p = α , + β , • · · · • (cid:63) · · · (cid:63)... ... . . . ... (cid:63) · · · (cid:63) , (184)where the lower-right ( p − × ( p −
1) star submatrices enjoy amazing structural properties. Atfirst, we need an:
Observation 10.9.
Let p (cid:62) be a positive integer, let A be a p × p matrix, let B be a p × pmatrix such that both its -st, ( p + -th rows are (1 , , . . . , (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) zeros ) . Then there holds: ( A B ) (cid:48) = A (cid:48)(cid:48) B (cid:48)(cid:48)(cid:48) , where ( A B ) (cid:48) means the ( p − × ( p − matrix obtained by deleting the first row and column ofA B, and where A (cid:48)(cid:48) means the ( p − × p − matrix obtained by deleting the first row and thecolumns , p + of A, and where B (cid:48)(cid:48)(cid:48) means the p − × ( p − matrix obtained by deleting thefirst column and the rows , p + of B. (cid:3) Now, noting that:
G X τ,ρ p = G (cid:0) X p I τ,ρ p (cid:1) = (cid:0) G X p (cid:1) I τ,ρ p , thanks to the above two observations, the ( p − × ( p −
1) star submatrices enjoy the forms: (cid:63) · · · (cid:63)... . . . ...(cid:63) · · · (cid:63) = X G p I τ − ,ρ − p − , (185)where X G p is the ( p − × p −
1) matrix obtained by deleting the first row and the columns 1 , p + G X p .Comparing (185) and (182), we immediately see that the star submatrices have the same struc-tures as X ,ν p , X τ,ρ p , which is the cornerstone of our induction approach.10.3. Study the morphism of left-multiplying by G . Let us denote by: D ( α , + β , ) ⊂ Mat p × p ( K )the Zariski open set where α , + β , (cid:44)
0. Now, consider the regular map of left-multiplying bythe function matrix G : L G : D ( α , + β , ) −→ D ( α , + β , ) X p (cid:55)−→ G X p . Of course, it is not surjective, as (184) shows that its image lies in the variety: ∩ pi = { α i , + β i , = } . n order to compensate this loss of surjectivity, combing with the regular map: e : D ( α , + β , ) −→ Mat ( p − × ( K ) X p (cid:55)−→ ( α , + β , | · · · | α p , + β p , ) T , we construct a regular map: L G ⊕ e : D ( α , + β , ) −→ (cid:18) ∩ p (cid:96) = { α i , + β i , = } ∩ D ( α , + β , ) (cid:19) ⊕ Mat ( p − × ( K ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : (cid:122) , which turns out to be an isomorphism. In fact, it has the inverse morphism: (cid:18) ∩ p (cid:96) = { α i , + β i , = } ∩ D ( α , + β , ) (cid:19) ⊕ Mat ( p − × ( K ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = (cid:122) −→ D ( α , + β , ) Y ⊕ ( s , . . . , s p ) T (cid:55)−→ − G · Y , where the matrix − G is the “inverse” of the regular function matrix G in (183): s α , + β , ... . . . s p α , + β , . (186)Now, let us denote by: π p : Mat p × p ( K ) −→ Mat ( p − × p − ( K )the projection map obtained by deleting the first row and the columns 1 , p +
1. Let us denote also: L G : = π p ◦ L G . It is worth to mention that there is a natural isomorphism: R : (cid:122) ∼ −−→ D ( α , + β , ) , Y ⊕ ( s , . . . , s p ) T (cid:55)−→ ?where ? is Y but replacing ( b , , . . . , b p , ) T by ( s , . . . , s p ) T , and thus we obtain a commutativediagram: D ( α , + β , ) L G ⊕ e (cid:47) (cid:47) L G (cid:41) (cid:41) (cid:122) π p ⊕ (cid:15) (cid:15) R (cid:47) (cid:47) D ( α , + β , ) π p (cid:117) (cid:117) Mat ( p − × p − ( K ) , (187)where the horizontal maps are isomorphisms, and where the right vertical map is surjective withfibre: ker π p (cid:124)(cid:123)(cid:122)(cid:125) K -linear space ∩ D ( α , + β , ) . Recalling the end of Subsection 10.2, we in fact received the following key observation. bservation 10.10. For every positive integer p (cid:62) , for every integer (cid:96) = · · · p − , the imageof the variety: (cid:96) X p ∩ D ( α , + β , ) ⊂ D ( α , + β , ) under the map: L G : D ( α , + β , ) −→ Mat ( p − × p − ( K ) is contained in the variety: (cid:96) X p − ⊂ Mat ( p − × p − ( K ) . (cid:3) A technical lemma.
Now, we carry out one preliminary lemma for the final proof of Propo-sition 10.4.For all positive integers p (cid:62)
3, for every integer (cid:96) = · · · p −
1, for every fixed ( p − × ( p − J of rank (cid:96) , denote the space which consists of all the p × p matrices of the form: z , z , · · · z , p z , ... Jz p , by J S p ,(cid:96) (cid:27) K p − . For every integer j = (cid:96), (cid:96) +
1, denote by J S jp ,(cid:96) ⊂ J S p ,(cid:96) the subvariety that consistsof all the matrices having rank (cid:54) j . Lemma 10.11.
The codimensions of J S jp ,(cid:96) are: codim J S jp ,(cid:96) = p − − (cid:96) ) + ( j = (cid:96) ) , p − − (cid:96) ( j = (cid:96) + . Proof. Step 1.
We claim that the codimensions of J S jp ,(cid:96) are independent of the matrix J .Indeed, choose two invertible ( p − × ( p −
1) matrices L and R , which normalize the matrix J by multiplications on both sides: L J R = . . . . . . = : J , where all the entries of J are zeros except the last (cid:96) copies of 1 in the diagonal. Therefore, weobtain an isomorphism: LR : J S p ,(cid:96) ∼ −−→ J S p ,(cid:96) S (cid:55)−→ (cid:32) L (cid:33) S (cid:32) R (cid:33) whose inverse is: L − R − : J S p ,(cid:96) ∼ −−→ J S p ,(cid:96) S (cid:55)−→ (cid:32) L − (cid:33) S (cid:32) R − (cid:33) . ince the map LR preserves the rank of matrices, it induces an isomorphism between J S jp ,(cid:96) and J S jp ,(cid:96) , which concludes the claim. Step 2.
For J , doing elementary row and column operations, we get: rank K z , z , · · · z , p z , ... J z p , = rank K z , z , · · · z , p − (cid:96) z , p − (cid:96) + · · · z , p z , ... z p − (cid:96), z p − (cid:96) + , ... . . . z p , = rank K z , − (cid:80) pk = p − (cid:96) + z k , z , k z , · · · z , p − (cid:96) · · · z , ... z p − (cid:96), ... . . . = rank K z , − (cid:80) pk = p − (cid:96) + z k , z , k z , · · · z , p − (cid:96) z , ... z p − (cid:96), + (cid:96). Step 3.
In the K -Euclidian space K N − with coordinates ( z , , z , , . . . , z , N , z , , . . . , z N , ), thealgebraic subvariety defined by the rank inequality: rank K z , − (cid:80) pk = p − (cid:96) + z k , z , k z , · · · z , p − (cid:96) z , ... z p − (cid:96), (cid:54) (cid:54) p − − (cid:96) ) + p − − (cid:96) ). (cid:3) Proof of Proposition 10.4.
Recalling the definition (155), and applying Lemma 10.3, wereceive:
Corollary 10.12.
For every integers p (cid:62) , the di ff erence of the varieties: p X p \ p − X p ⊂ Mat p × p ( K ) s exactly the quasi-variety: (cid:110) α + · · · + α p + β + · · · + β p = (cid:111)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = p ∩ (cid:110) rank K { α , . . . , α p } = p (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ⇔ det ( α |···| α ) (cid:44) (cid:111) , whose codimension is p.Proof. For every p × p matrix: p X p \ p − X p (cid:51) X p = ( α , . . . , α p , β , . . . , β p ) , applying now Lemma 10.3 to condition (157): rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α p , α ν + ( β + · · · + β p ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : β (cid:9) (cid:54) p − ( ν = ··· p ) , since: rank K (cid:8) α , . . . , α p (cid:9) = p , we immediately receive: α + · · · + α p + β + · · · + β p = . On the other hand, for all matrices: X p = ( α , . . . , α p , β , . . . , β p )satisfying the above identity, (ii) holds immediately. Noting that the p × p matrix in (158) has avanishing sum of all its columns, it has rank (cid:54) p −
1, i.e. (iii) holds too. (cid:3)
Now, we give a complete proof of the Codimension Induction Formulas.
Proof of (168) . This is a direct consequence of the above corollary. (cid:3)
Proof of (169) . By Observation 10.10, under the map: L G : D ( α , + β , ) −→ Mat ( p − × p − ( K ) , the image of the variety: p − X p ∩ D ( α , + β , )is contained in the variety: p − X p − ⊂ Mat ( p − × p − ( K ) . Now, let us decompose the variety p − X p − into three pieces: p − X p − = p − X p − ∪ (cid:0) p − X p − \ p − X p − (cid:1) ∪ (cid:0) p − X p − \ p − X p − (cid:1) , (188)where each matrix ( α , . . . , α p − , β , . . . , β p − ) in the first (resp. second , third ) piece satisfies: rank K ( α , . . . , α p − ) (cid:54) p − (resp. = p − , = p − ) . (189)Pulling back (188) by the map L G , we see that: p − X p ∩ D ( α , + β , )is contained in: L − G ( p − X p − ) ∪ L − G (cid:0) p − X p − \ p − X p − (cid:1) ∪ L − G (cid:0) p − X p − \ p − X p − (cid:1) . (190)Firstly, for every point in the first piece: Y ∈ p − X p − , hanks to the commutative diagram (187), we receive the fibre dimension: dim L − G ( Y ) = dim ker π p ∩ D ( α , + β , ) = dim Mat p × p ( K ) − dim Mat ( p − × p − ( K ) . Now, applying Corollary 8.3 to the regular map L restricted on: L − G ( p − X p − ) ⊂ Mat p × p ( K )we receive the codimension estimate: codim L − G ( p − X p − ) (cid:62) codim p − X p − (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) p − C p − . (191)Secondly, for every point in the second piece: Y ∈ p − X p − \ p − X p − , to look at the fibre of L − G ( Y ), thanks to the commutative diagram (187), we can use: L − G = (cid:0) R ◦ ( L G ⊕ e ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) an isomorphism (cid:1) − ◦ π − p , (192)and obtain: L − G ( Y ) ∩ (cid:0) p − X p ∩ D ( α , + β , ) (cid:1) (cid:27) R ◦ ( L G ⊕ e ) L − G ( Y ) ∩ R ◦ ( L G ⊕ e ) (cid:0) p − X p ∩ D ( α , + β , ) (cid:1) (cid:27) π − p ( Y ) ∩ R ◦ ( L G ⊕ e ) (cid:0) p − X p ∩ D ( α , + β , ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : ♣ [use (192) ] . Observe now that every matrix: ( α | · · · | α p | β | · · · | β p ) ∈ ♣ satisfies the rank estimate: rank K (cid:0) α | · · · | α p (cid:1) (cid:54) p − . Moreover, noting that the lower-right ( p − × ( p −
1) submatrix J of (cid:0) α | · · · | α p (cid:1) — which isthe left ( p − × ( p −
1) submatrix of Y — has rank: rank K J = p − [see (189) ] , by applying Lemma 10.11, we get that: ♣ ⊂ π − p ( Y )has codimension greater or equal to: codim J S p − p , p − = p − − ( p − = . In other words: L − G ( Y ) ∩ (cid:0) p − X p ∩ D ( α , + β , ) (cid:1) ⊂ L − G ( Y )has codimension (cid:62)
1. Thus, applying Corollary 8.3 to the map L G restricted on: L − G (cid:0) p − X p − \ p − X p − (cid:1) ∩ (cid:0) p − X p ∩ D ( α , + β , ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : II ⊂ Mat p × p ( K ) , e receive the codimension estimate: codim II (cid:62) codim (cid:0) p − X p − \ p − X p − (cid:1) + (cid:62) codim p − X p − + (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) p − C p − + . (193)Thirdly, for every point in the third piece: Y ∈ p − X p − \ p − X p − , thanks to the diagram (187): L − G = ( L G ⊕ e (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) (cid:27) ) − ◦ ( π p ⊕ ) − , (194)we receive: L − G ( Y ) ∩ (cid:0) p − X p ∩ D ( α , + β , ) (cid:1) (cid:27) ( L G ⊕ e ) L − G ( Y ) ∩ ( L G ⊕ e ) (cid:0) p − X p ∩ D ( α , + β , ) (cid:1) (cid:27) ( π p ⊕ ) − ( Y ) ∩ ( L G ⊕ e ) (cid:0) p − X p ∩ D ( α , + β , ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : ♠ [use (194) ] . Recalling Corollary 10.12, the sum of all columns of Y — the bottom ( p −
1) rows of ( α | · · · | α p | β | · · · | β p ) — is zero. Thus, every element:( α | · · · | α p | β | · · · | β p ) ⊕ ( s , . . . , s p ) T ∈ ♠ not only satisfies: rank K ( α | · · · | α p ) (cid:54) p − , (195)but also satisfies: α + · · · + α p + β + · · · + β p = ( α , + · · · + α , p + β , + · · · + β , p (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) only this first entry could be nonzero , , . . . , (cid:124) (cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32) (cid:125) ( p −
1) copies ) T . Remembering that: α + β = ( α , + β , , , . . . , (cid:124) (cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32) (cid:125) ( p −
1) copies ) T , summing the above two identities immediately yields: α + · · · + α p + β + · · · + β p = α , + · · · + α , p + β , + · · · + β , p , , . . . , (cid:124) (cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32) (cid:125) ( p −
1) copies ) T . (196)Now, note that (157) (‘matrices ranks’) in condition (ii) are preserved under the map L G (‘Gaussianeliminations’), in particular, for ν =
1, the image satisfies: rank K (cid:8) α + ( β + · · · + β p ) , α , . . . , α p (cid:9) (cid:54) p − , which, by adding the column vectors 2 · · · p to the first one, is equivalent to: rank K (cid:8) α + · · · + α p + β + · · · + β p , α . . . , α p (cid:9) (cid:54) p − . Remember (196), and recalling Corollary 10.12:the bottom ( p − × ( p −
1) submatrix of ( α | · · · | α p ) is of full rank ( p − , (197) e immediately receive: α , + · · · + α , p + β , + · · · + β , p = (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) codim = . Therefore, by applying Lemma 10.11, the restrictions (195) and (197) contribute one extra codi-mension: codim J S p − p , p − = . Thus, we see that ‘the fibre in fibre’:( π p ⊕ ) − ( Y ) ∩ ♠ ⊂ ( π p ⊕ ) − ( Y )has codimension greater or equal to: 1 + = . Now, applying once again Corollary 8.3 to the map L G restricted on: L − G (cid:0) p − X p − \ p − X p − (cid:1) ∩ (cid:0) p − X p ∩ D ( α , + β , ) (cid:1)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = : III ⊂ Mat p × p ( K ) , we receive the codimension estimate: codim III (cid:62) codim (cid:0) p − X p − \ p − X p − (cid:1) + (cid:62) codim p − X p − + (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) p − C p − + . (198)Summarizing (190), (191), (193), (198), we receive the codimension estimate: codim p − X p ∩ D ( α , + β , ) (cid:62) min (cid:8) codim p − X p − , codim p − X p − + , codim p − X p − + (cid:9) . By permuting the indices, we know that all: p − X p ∩ D ( α i , + β i , ) ⊂ Mat p × p ( K ) ( i = ··· p ) have the same codimension, and so does their union: p − X p ∩ D ( α + β ) = ∪ pi = (cid:0) p − X p ∩ D ( α i , + β i , ) (cid:1) ⊂ Mat p × p ( K ) . Finally, taking codimension on both sides of: p − X p = (cid:0) p − X p ∩ V ( α + β ) (cid:1) ∪ (cid:0) p − X p ∩ D ( α + β ) (cid:1) , Proposition 10.2 and the preceding estimate conclude the proof. (cid:3)
Proof of (170) . If (cid:96) (cid:62)
2, decompose the variety (cid:96) X p − into three pieces: (cid:96) X p − = (cid:96) − X p − ∪ (cid:0) (cid:96) − X p − \ (cid:96) − X p − (cid:1) ∪ (cid:0) (cid:96) X p − \ (cid:96) − X p − (cid:1) ;and if (cid:96) =
1, decompose the variety (cid:96) X p − into two pieces: (cid:96) X p − = (cid:96) − X p − ∪ (cid:0) (cid:96) X p − \ (cid:96) − X p − (cid:1) . Now, by mimicking the preceding proof, namely by applying Lemma 10.11 and Corollary 8.3,everything goes on smoothly with much less e ff ort, because there is no need to perform delicatecodimension estimates such as (198). (cid:3) Proof of Core Lemma 9.5. If N =
1, there is nothing to prove. Assume now N (cid:62) π c + r , N : Mat (2 c + r ) × N + ( K ) −→ Mat N × N ( K ) (cid:0) α , . . . , α p , β , . . . , β p (cid:1) (cid:55)−→ (cid:0)(cid:98) α , . . . , (cid:98) α p , (cid:98) β , . . . , (cid:98) β p (cid:1) , where each widehat vector is obtained by extracting the first N rows (entries) out of the original2 c + r rows (entries).Now, for every point:( α , . . . , α p , β , . . . , β p ) ∈ M N c + r ⊂ Mat (2 c + r ) × N + ( K ) , in restriction (148), by setting ν =
0, we receive: rank K (cid:8) α , . . . , α N , α + ( β + β + · · · + β N ) (cid:9) (cid:54) N − . Dropping the last column and extracting the first N rows, we get: rank K (cid:8)(cid:98) α , . . . , (cid:98) α N (cid:9) (cid:54) N − . Similarly, in restriction (149), by dropping the first column and extracting the first N rows, for all τ = · · · N − ρ = τ + · · · N , we obtain: rank K (cid:8)(cid:98) α + (cid:98) β , . . . , (cid:98) α τ + (cid:98) β τ , (cid:98) α τ + , . . . (cid:98) α ρ . . . , (cid:98) α N , (cid:98) α ρ + ( (cid:98) β τ + + · · · + (cid:98) β N ) (cid:9) (cid:54) N − , where we omit the column vector (cid:98) α ρ in the box. Summarizing the above two inequalities, ( (cid:98) α , . . . , (cid:98) α p , (cid:98) β , . . . , (cid:98) β p )satisfies the restriction (156) – (158): π c + r , N ( α , . . . , α p , β , . . . , β p ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = ( (cid:98) α ,..., (cid:98) α p , (cid:98) β ,..., (cid:98) β p ) ∈ N − X N ⊂ Mat N × N ( K ) . Therefore: π c + r , N ( M N c + r ) ⊂ N − X N . Moreover, for every point Y ∈ N − X N , the ‘fibre in fibre’: π − c + r , N ( Y ) ∩ M N c + r ⊂ π − c + r , N ( Y ) , thanks to (147), has codimension (cid:62) c + r . Thus a direct application of Corollary 8.3 yields: codim M N c + r (cid:62) codim N − X N + c + r [use (180) ] (cid:62) N + + c + r . Repeating the same reasoning, we obtain: codim M N − η c + r (cid:62) codim N − η − X N − η + c + r [use (180) ] (cid:62) N − η + + c + r . Remembering 2 c + r (cid:62) N , we conclude the proof. (cid:3) ‘Macaulay2’, ‘Maple’ et al. vs. the Core Lemma. Believe it or not, concerning the CoreLemma or the Core Codimension Formulas, ‘Macaulay2’ – a professional software system devotedto supporting research in algebraic geometry and commutative algebra – is not strong enough tocompute the precise codimensions of the involved determinantal ideals, even in small dimensions p (cid:62)
4. And unfortunately, so do other mathematical softwares, like ‘Maple’...This might indicate some weaknesses of current computers. Since the Core Lemma or a variationof it should be a crucial step in the constructions of ample examples , the dream of finding explicitexamples with rational coe ffi cients, firstly in small dimensional cases, could be kind of a challengefor a moment. 11. A rough estimate of lower degree bound E ff ective results. Recalling Subsection 5.3, we first provide an e ff ective Theorem 11.1.
For all N (cid:62) , for any (cid:15) , . . . , (cid:15) c + r ∈ { , } , Theorem 5.2 holds for (cid:114) = . and forall d (cid:62) N N / − .Proof. Setting δ c + r + = [see (97) ] ( N + µ N , N (cid:54) N N / − . For the sake of completeness, we present all computational details in Subsection 11.2 below. (cid:3)
Hence, the product coup in Subsection 5.3 yields
Theorem 11.2.
In Theorem 5.1, for (cid:114) = , the lower bound d ( − = N N works. (cid:3) Computational details.
We specify (94) – (97) as follows. Recalling that δ c + r + = (cid:114) =
1, for every integer l = c + r + · · · N , we choose: µ l , = l δ l + l + , (199)and inductively we choose: µ l , k = k − (cid:88) j = l µ l , j + ( l − k ) δ l + l + ( k = ··· l ) . (200)Actually, we take the above values in purpose, because they also work in the degree estimates inour coming paper.For every integer l = c + r + · · · N , for every integer k = · · · l , let: S l , k : = k (cid:88) j = µ l , j . (201)For k = · · · l , we have: [see (200) ] S l , k − S l , k − (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) = µ l , k = l S l , k − + ( l − k ) δ l + l + . Moving the term ‘ − S l , k − ’ to the right hand side, we receive: S l , k = ( l + S l , k − + ( l − k ) δ l + l + . ividing by ( l + k on both sides, we receive: S l , k ( l + k = S l , k − ( l + k − + (cid:16) l δ l + l + (cid:17) l + k − δ l k ( l + k . Noting that the two underlined terms have the same structure, doing induction backwards k · · · S l , k ( l + k = S l , ( l + + (cid:16) l δ l + l + (cid:17) k (cid:88) j = l + j − δ l k (cid:88) j = j ( l + j . Now, applying the following two elementary identities: k (cid:88) j = l + j = l (cid:16) − l + k (cid:17) , k (cid:88) j = j ( l + j = l + l (cid:16) + k ( l + k + − + k ( l + k (cid:17) , and recalling (199): S l , = µ l , = l δ l + l + , we obtain: S l , k ( l + k = l δ l + l + + (cid:16) l δ l + l + (cid:17) l (cid:16) − l + k (cid:17) − δ l l + l (cid:16) + k ( l + k + − + k ( l + k (cid:17) . Next, multiplying by ( l + k on both sides, we get: S l , k = (cid:16) l δ l + l + (cid:17) (cid:16) ( l + k + ( l + k l − l (cid:17) − δ l l (cid:16) ( l + k + + k − (1 + k ) ( l + (cid:17) = (cid:16) l δ l + l + (cid:17) (cid:16) ( l + k + l − l (cid:17) − δ l l (cid:16) ( l + k + + k − (1 + k ) ( l + (cid:17) . (202)Recalling (96), we have: δ l + = l µ l , l [use (200) for k = l ] = l (cid:16) l − (cid:88) j = l µ l , j + l + (cid:17) [use (201) for k = l − ] = l S l , l − + l (4 l + [use (202) for k = l − ] = (cid:16) l δ l + l + (cid:17) (cid:16) l ( l + l − l (cid:17) − δ l (cid:16) ( l + l + l − − l ( l + (cid:17) + l (4 l + = δ l (cid:16) l ( l + l − ( l + l + (cid:17)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:62) + (4 l + l ( l + l . (203)Throwing away the first positive part, we receive the estimate: δ l + > (4 l + l ( l + l . (204) herefore, for all l (cid:62) c + r +
2, we have the estimate of (203): δ l + = l ( l + l δ l − (cid:16) ( l + l − (cid:17) δ l + (4 l + l ( l + l [use (204) ] < l ( l + l δ l − (cid:16) ( l + l − (cid:17) (cid:0) l − + (cid:1) ( l − l l − + (4 l + l ( l + l = l ( l + l δ l − (cid:104)(cid:16) ( l + l − (cid:17) ( l − l l − − l ( l + l (cid:105) − (cid:104)(cid:16) ( l + l − (cid:17) ( l − l l − − l ( l + l (cid:105) . Since 2 c + r (cid:62) N (cid:62) c , r cannot be both zero, hence l (cid:62) c + r + (cid:62) (cid:104)(cid:16) ( l + l − (cid:17) ( l − l l − − l ( l + l (cid:105) = l ( l + l (cid:34)(cid:16) − l + l (cid:17) ( l − l l − − (cid:35) (cid:62) l ( l + l (cid:34)(cid:16) − + (cid:17) (3 − − − (cid:35) > , and that the second underlined bracket is also positive: (cid:104)(cid:16) ( l + l − (cid:17) ( l − l l − − l ( l + l (cid:105) = l ( l + l (cid:34)(cid:16) − l + l (cid:17) ( l − l l − − (cid:35) (cid:62) l ( l + l (cid:34)(cid:16) − + (cid:17) (3 −
1) 3 − − (cid:35) > . Consequently, we have the neat estimate suitable for the induction: δ l + (cid:54) l ( l + l δ l ( l = c + r + ··· N − , (205)which for convenience, we may assume to be satisfied for l = N by just defining δ N + : = N µ N , N .In fact, using these estimates iteratively, we may proceed as follows:( N + µ N , N = N + N N µ N , N (cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) = δ N + = N + N δ N + (206) [use (205) ] < N + N δ c + r + N (cid:89) l = c + r + l ( l + l . Noting that (203) yields: δ c + r + = (cid:104) δ l (cid:16) l ( l + l − ( l + l + (cid:17) + (4 l + l ( l + l (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = c + r + [recall δ c + r + = ] < l ( l + l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = c + r + , hus the above two estimates yield:( N + µ N , N < N + N N (cid:89) l = c + r + l ( l + l . (207)For the convenience of later integration, we prefer the term ( l + l + to ( l + l , therefore we firstlytransform: N (cid:89) l = c + r + l ( l + l = N (cid:89) l = c + r + ll + l ( l + l + = N (cid:89) l = c + r + ll + N (cid:89) l = c + r + l N (cid:89) l = c + r + ( l + l + = c + r + N + N (cid:89) l = c + r + l N (cid:89) l = c + r + ( l + l + , whence (207) becomes: ( N + µ N , N < c + r + N N (cid:89) l = c + r + l N (cid:89) l = c + r + ( l + l + [recall c + r (cid:54) N − ] (cid:54) N (cid:89) l = c + r + l N (cid:89) l = c + r + ( l + l + . (208)Now, we estimate the dominant term: N (cid:89) l = c + r + l N (cid:89) l = c + r + ( l + l + . as follows.Remembering 2 c + r (cid:62) N , we receive: c + r (cid:62) (2 c + r ) / (cid:62) N / , and hence for N (cid:62) N (cid:89) l = c + r + l = ln N + N − (cid:88) l = c + r + ln l < ln N + (cid:90) Nc + r + ln x dx (cid:54) ln N + (cid:90) NN / + ln x dx = ln N + ( x ln x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) NN / + . imilarly, when N (cid:62)
4, noting that N (cid:62) N / +
2, we get the estimate:ln N (cid:89) l = c + r + ( l + l + = ( N +
1) ln ( N + + N ln N + N − (cid:88) l = c + r + l ln l (cid:54) ( N +
1) ln ( N + + N ln N + (cid:90) NN / + x ln x dx = ( N +
1) ln ( N + + N ln N + (cid:16) x ln x − x (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) NN / + . Summing the above two estimates, for N (cid:62) N (cid:89) l = c + r + l + ln N (cid:89) l = c + r + ( l + l + (cid:54) ln N + ( x ln x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) NN / + + ( N +
1) ln ( N + + N ln N + (cid:16) x ln x − x (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) NN / + = N ln N −
12 ( N / + ln ( N / + − (cid:16) N − (cid:17) − ( N / +
1) ln ( N / + ++ ( N +
1) ln ( N + + (2 N +
1) ln N (209) = N ln N − O ( N ) , as N → ∞ . (210)In order to have a neat lower bound, we would like to have:( N + µ N , N (cid:54) N N / − . (211)In fact, using the estimates (208), (209), when N (cid:62)
48, we can show by hand that (211) holdstrue. For N = · · ·
47, we can use a mathematical software ‘Maple’ to check the above estimate.Finally, for N = · · ·
13, we ask ‘Maple’ to compute δ N + explicitly, and thereby, thanks to (206),we again prove the estimate (211).12. Some Improvements of MCM
General core codimension formulas.
In order to lower the degree bound d of MCM, wewill modify the hypersurface constructions. Of course, we would like to reduce the number ofmoving coe ffi cient terms, and this will be based on the General Core Lemma 12.6 below.For every integers p (cid:62) q (cid:62)
2, for every integer 0 (cid:54) (cid:96) (cid:54) q , we first estimate the codimension (cid:96) C p , q of the algebraic variety: (cid:96) X p , q ⊂ Mat p × q ( K )which consists of p × q matrices X p , q = ( α , . . . , α q , β , . . . , β q ) such that: (i) the first q column vectors have rank: rank K (cid:8) α , . . . , α q (cid:9) (cid:54) (cid:96) ; (ii) for every index ν = · · · q , replacing α ν with α ν + ( β + · · · + β q ) in the collection of columnvectors { α , . . . , α q } , there holds the rank inequality: rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α q , α ν + ( β + · · · + β q ) (cid:9) (cid:54) p − iii) for every integer τ = · · · q −
1, for every index ρ = τ + · · · q , replacing α ρ with α ρ + ( β τ + + · · · + β q ) in the collection of column vectors { α + β , . . . , α τ + β τ , α τ + , . . . , α ρ , . . . , α q } , thereholds the rank inequality: rank K (cid:8) α + β , . . . , α τ + β τ , α τ + , . . . , (cid:98) α ρ , . . . , α q , α ρ + ( β τ + + · · · + β q ) (cid:9) (cid:54) q − . Repeating the same reasoning as in Section 10, we may proceed as follows. Firstly, here is avery analogue of Proposition 10.1:
Proposition 12.1.
For every integers p (cid:62) q (cid:62) , the codimension value (cid:96) C p , q for (cid:96) = is: C p , q = p q + p − q + . (cid:3) Next, we obtain an analogue of Proposition 10.2:
Proposition 12.2.
For every integers p (cid:62) q (cid:62) , the codimensions (cid:96) C p , q of the algebraic varieties: { α + β = } ∩ (cid:96) X p , q ⊂ Mat p × q ( K ) read according to the values of (cid:96) as: (cid:96) C p , q = p + min { p − q + , p } ( (cid:96) = q ) , p + p − q + ( (cid:96) = q − , p + ( p − (cid:96) ) ( q − (cid:96) ) ( (cid:96) = ··· q − . The last two lines are easy to obtain, while the first line is a consequence of Lemma 10.3. (cid:3)
Now, we deduce the analogue of Proposition 10.4:
Proposition 12.3 ( General Codimension Induction Formulas). (i)
For every positive integersp (cid:62) q (cid:62) , for (cid:96) = q, the codimension value q C p , q satisfies: q C p , q = min (cid:8) p , q − C p , q (cid:9) . (ii) For every positive integers p (cid:62) q (cid:62) , for (cid:96) = q − , the codimension value (cid:96) C p , q satisfies: q − C p , q (cid:62) min (cid:8) q − C p , q , q − C p − , q − + p − q + , q − C p − , q − + , q − C p − , q − (cid:9) . (iii) For all positive integers p (cid:62) q (cid:62) , for all integers (cid:96) = · · · q − , the codimension values (cid:96) C p , q satisfy: (cid:96) C p , q (cid:62) min (cid:8) (cid:96) C p , q , (cid:96) C p − , q − + ( p − (cid:96) ) + ( q − (cid:96) ) − , (cid:96) − C p − , q − + ( q − (cid:96) ) , (cid:96) − C p − , q − (cid:9) . (cid:3) Similar to Proposition 10.5, we have:
Proposition 12.4.
For the initial cases p (cid:62) q = , there hold the codimension values: C p , = p − , C p , = p − , C p , = p . (cid:3) Finally, by the same induction proof as in Proposition 10.6, we get:
Proposition 12.5 ( General Core Codimension Formulas).
For all integers p (cid:62) q (cid:62) , there holdthe codimension estimates: (cid:96) C p , q (cid:62) ( p − (cid:96) ) ( q − (cid:96) ) + p − q + l + ( (cid:96) = ··· q − , and the core codimension identity: q C p , q = p . (cid:3) Actually, we could prove that the above estimates are identities, yet it is not really necessary. General Core Lemma.
Similar to (146), for every integer k = · · · N −
1, we introduce thealgebraic subvariety: M N , k c + r ⊂ Mat (2 c + r ) × ( N + + k + ( K )consisting of all ( c + r + c ) × ( N + + k +
1) matrices ( α | α | · · · | α N | β | β | · · · | β k ) such that: (i) the sum of these ( N + + k +
1) colums is zero: α + α + · · · + α N + β + β + · · · + β k = ; (212) (ii) for every index ν = · · · k , replacing α ν with α ν + ( β + β + · · · + β k ) in the collection ofcolumn vectors { α , α , . . . , α N } , there holds the rank inequality: rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α N , α ν + ( β + β + · · · + β k ) (cid:9) (cid:54) N − (iii) for every integer τ = · · · k −
1, for every index ρ = τ + · · · k , replacing α ρ with α ρ + ( β τ + + · · · + β k ) in the collection of column vectors { α + β , . . . , α τ + β τ , α τ + , . . . , α ρ , . . . , α N } , thereholds the rank inequality: rank K (cid:8) α + β , α + β , . . . , α τ + β τ , α τ + , . . . , (cid:98) α ρ , . . . , α N , α ρ + ( β τ + + · · · + β k ) (cid:9) (cid:54) N − . Lemma 12.6 ( Sharp Core Lemma of MCM).
For every positive integers N (cid:62) , for every in-tegers c , r (cid:62) with c + r (cid:62) N, for every integer k = · · · N − , there holds the codimensionestimate: codim M N , k c + r (cid:62) k + C c + r − N + k + , k + + (2 c + r ) (cid:62) c + r ) − N + k + . The term (2 c + r ) comes from (212). When k = N −
1, there is nothing to prove. When k < N −
1, noting that all matrices in (ii) and (iii) have the same last column α N , we may doGaussian eliminations with respect to this column, and then by much the same argument as before,we receive the estimate. (cid:3) Actually, these two estimates are identities.12.3.
Minimum necessary number of moving coe ffi cient terms. Firstly, letting: codim M N , k c + r (cid:62) dim P ◦ (T P N ) = N − , we receive the lower bound: k (cid:62) N − c + r ) − , which indicates that at the step N , the least number of moving coe ffi cient terms, if necessary,should be: 3 N − c + r ) − + . When 3 N − c + r ) − (cid:54)
0, no moving coe ffi cient terms are needed, thanks to the following: Lemma 12.7. (Elementary Core Lemma)
Let: M N , − c + r ⊂ Mat (2 c + r ) × ( N + ( K ) consist of all ( c + r + c ) × ( N + matrices ( α | α | · · · | α N ) such that: (i) the sum of these ( N + colums is zero: α + α + · · · + α N = ; (213) ii) there holds the rank inequality: rank K (cid:8) α , . . . , (cid:98) α ν , . . . , α N (cid:9) (cid:54) N − ( ν = ··· N ) . Then one has the codimension identity: codim M N , − c + r = c + r ) − N + . (cid:3) Next, for η = · · · n −
1, in the step N − η , letting: codim M N − η, k c + r (cid:62) dim v ,..., v η P ◦ (T P N ) = N − η − , we receive: 2 (2 c + r ) − ( N − η ) + k + (cid:62) N − η − , that is: k (cid:62) N − c + r ) − − η, which indicates that, at the step N − η , the least number of moving coe ffi cient terms, if necessary,should be: 3 N − c + r ) − − η + . When 3 N − c + r ) − − η (cid:54)
0, no moving coe ffi cient terms are needed, thanks to the ElementaryCore Lemma.12.4. Improved Algorithm of MCM.
When 3 N − c + r ) − >
0, in order to lower the degrees,we improve the hypersurface equations (92) as follows.Firstly, when 3 N − c + r ) − = p is even, the following hypersurface equations are suitablefor MCM: F i = N (cid:88) j = A ji z dj + p − (cid:88) η = (cid:88) (cid:54) j < ··· < j N − η (cid:54) N p − η (cid:88) k = M j ,..., j N − η ; j k i z µ N − η, k j · · · (cid:91) z µ N − η, k j k · · · z µ N − η, k j p − η z d − (2 p − η ) µ N − η, k − N + η − p ) j k z j p − η + · · · z j N − η . (214)Secondly, when 3 N − c + r ) − = p + F i = N (cid:88) j = A ji z dj + p (cid:88) η = (cid:88) (cid:54) j < ··· < j N − η (cid:54) N p + − η (cid:88) k = M j ,..., j N − η ; j k i z µ N − η, k j · · · (cid:91) z µ N − η, k j k · · · z µ N − η, k j p + − η z d − (2 p + − η ) µ N − η, k − N + η − p − j k z j p − η + · · · z j N − η . (215)Of course, all integers µ • , • and the degree d are to be determined by some improved Algorithm ,so that all the obtained symmetric forms are negatively twisted. And then we may estimate thelower bound d accordingly. We leave this standard process to the interested reader. Why is the lower degree bound d so large in MCM. Because we could not enter theintrinsic di ffi culties, firstly of solving some huge linear systems to obtain su ffi ciently many (nega-tively twisted, large degree) symmetric di ff erential forms (see [7, Theorem 2.7]), and secondly ofproving that the obtained symmetric forms have discrete base locus. What we have done is onlyfocusing on the extrinsic negatively twisted symmetric forms with degrees (cid:54) n , obtained by someminors of the hypersurface equations (cid:14) di ff erentials matrix.Our tool is coarse, based on some robust extrinsic geometric (cid:14) algebraic structures, yet our goalis delicate, to certify the conjectured intrinsic ampleness. So a large lower degree bound d (cid:29) Uniform Very-Ampleness of
Sym κ Ω X A reminder.
In [34], Fujita proposed the famous:
Conjecture 13.1. (Fujita)
Let M be an n -dimensional complex manifold with canonical line bun-dle K . If L is any positive holomorphic line bundle on M , then: (i) for every integer m (cid:62) n +
1, the line bundle L ⊗ m ⊗ K should be globally generated; (ii) for every integer m (cid:62) n +
2, the line bundle L ⊗ m ⊗ K should be very ample.Recall that, given a complex manifold X having ample cotangent bundle Ω X , the projectivizedtangent bundle P (T X ) is equipped with the ample Serre line bundle O P (T X ) (1). Denoting n : = dim X ,one has: dim P (T X ) = n − . Anticipating, we will show in Corollary 13.3 below that the canonical bundle of P (T X ) is: K P (T X ) (cid:27) O P (T X ) ( − n ) ⊗ π ∗ K X ⊗ , where π : P (T X ) → X is the canonical projection. Thus, for the complex manifold P (T X ) and theample Serre line bundle O P (T X ) (1), the Fujita Conjecture implies: (i) for every integer m (cid:62) n , the line bundle O P (T X ) ( m − n ) ⊗ π ∗ K X ⊗ is globally generated; (ii) for every integer m (cid:62) n +
1, the line bundle O P (T X ) ( m − n ) ⊗ π ∗ K X ⊗ is very ample.In other words, we receive the following by-products of the Fujita Conjecture. A Consequence of the Fujita Conjecture.
For any n-dimensional complex manifold X havingample cotangent bundle Ω X , there holds: (i) for every integer m (cid:62) n, the twisted m-symmetric cotangent bundle Sym m Ω X ⊗ K ⊗ X isglobally generated; (ii) for every integer m (cid:62) n + , the twisted m-symmetric cotangent bundle Sym m Ω X ⊗ K ⊗ X isvery ample. The canonical bundle of a projectivized vector bundle.
In this subsection, we recall someclassical results in algebraic geometry.Let X be an n -dimensional complex manifold, and let E be a holomorphic vector bundle on X having rank e . Let P ( E ) be the projectivization of E . Now, we compute its canonical bundle K P ( E ) as follows.Let π be the canonical projection: π : P ( E ) −→ X . irst, recall the exact sequence which defines the relative tangent bundle T π :0 −→ T π −→ T P ( E ) −→ π ∗ T X −→ , (216)and recall also the well known Euler exact sequence:0 −→ O P ( E ) −→ O P ( E ) (1) ⊗ π ∗ E −→ T π −→ . (217)Next, taking wedge products, the exact sequence (216) yields: ∧ n + e − T P ( E ) (cid:27) ∧ e − T π ⊗ π ∗ ∧ n T X , (218)and the Euler exact sequence (217) yields: O P ( E ) ( e ) ⊗ π ∗ ∧ e E (cid:27) O P ( E ) ⊗ ∧ e − T π (cid:27) ∧ e − T π . (219)Thus, we may compute the canonical line bundle as: K P ( E ) = ∧ n + e − Ω P ( E ) (cid:27) (cid:0) ∧ n + e − T P ( E ) (cid:1) ∨ [use the dual of (218) ] (cid:27) (cid:0) ∧ e − T π (cid:1) ∨ ⊗ (cid:0) π ∗ ∧ n T X (cid:1) ∨ [use the dual of (219) ] (cid:27) (cid:0) O P ( E ) ( e ) (cid:1) ∨ ⊗ (cid:0) π ∗ ∧ e E (cid:1) ∨ ⊗ (cid:0) π ∗ ∧ n T X (cid:1) ∨ (cid:27) O P ( E ) ( − e ) ⊗ π ∗ ∧ e E ∨ ⊗ π ∗ ∧ n Ω X (cid:27) O P ( E ) ( − e ) ⊗ π ∗ ∧ e E ∨ ⊗ π ∗ K X , where K X is the canonical line bundle of X . Proposition 13.2.
The canonical line bundle K P ( E ) of P ( E ) satisfies the formula: K P ( E ) (cid:27) O P ( E ) ( − e ) ⊗ π ∗ ∧ e E ∨ ⊗ π ∗ K X . (cid:3) In applications, first, we are interested in the case where E is the tangent bundle T X of X . Corollary 13.3.
One has the formula: K P (T X ) (cid:27) O P (T X ) ( − n ) ⊗ π ∗ K X ⊗ . (cid:3) More generally, we are interested in the case where X ⊂ V for some complex manifold V ofdimension n + r , and E = T V (cid:12)(cid:12)(cid:12) X . Corollary 13.4.
One has: K P (T V | X ) (cid:27) O P (T V | X ) ( − n − r ) ⊗ π ∗ K V (cid:12)(cid:12)(cid:12) X ⊗ π ∗ K X . (cid:3) In our applications, X , V are some smooth complete intersections in P N C , so their canonical linebundles K X , K V have neat expressions by the following classical theorem, whose proof is basedon the Adjunction Formula. Theorem 13.5.
For a smooth complete intersection:Y : = D ∩ · · · ∩ D k ⊂ P N C with divisor degrees: deg D i = d i ( i = ··· k ) , the canonical line bundle K X of X is: K X (cid:27) O X (cid:16) − N − + k (cid:88) i = d i (cid:17) . (cid:3) Proof of the Very-Ampleness Theorem 1.4.
Assume for the moment that the ambient field K = C . Recall that in our Ampleness Theorem 1.3, V = H ∩ · · · ∩ H c and X = H ∩ · · · ∩ H c + r with dim C X = n = N − ( c + r ). Then the above Corollary 13.4 and Theorem 13.5 imply: K P (T V | X ) (cid:27) O P (T V | X ) ( − n − r ) ⊗ π ∗ O P N K (cid:16) − N + + c (cid:88) i = d i + c + r (cid:88) i = d i (cid:17) . Also, recalling Theorem 5.1 and Proposition 4.4, for generic choices of H , . . . , H c + r , for anypositive integers a > b (cid:62)
1, the negatively twisted line bundle below is ample: O P (T V | X ) ( a ) ⊗ π ∗ O P N K ( − b ) . Recall the Fujita Conjecture that, by subsequent works of Demailly, Siu et al. (cf. the sur-vey [21]), it is known that L ⊗ m ⊗ K ⊗ is very ample for all large m (cid:62) + (cid:16) n + n (cid:17) . Consequently,the line bundle: O P (T V | X ) (cid:16) m a − n − r (cid:17) ⊗ π ∗ O P N K (cid:16) − m b − N + + c (cid:88) i = d i + c + r (cid:88) i = d i (cid:17) (220)is very ample.Also note that, for similar reason as the ampleness of (24), for all large integers (cid:96) (cid:62) (cid:96) ( N ): O P (T P N K ) (1) ⊗ π ∗ O P N K ( (cid:96) )is very ample. Consequently, so is: O P (T V | X ) (1) ⊗ π ∗ O P N K ( (cid:96) ) . (221)Now, recall the following two facts: (A) if O P (T V | X ) ( κ ) ⊗ π ∗ O P N K ( (cid:63) ) is very ample, then for every (cid:63) (cid:48) (cid:62) (cid:63) , O P (T V | X ) ( κ ) ⊗ π ∗ O P N K ( (cid:63) (cid:48) )is also very ample; (B) the tensor product of any two very ample line bundles remains very ample.Therefore, thanks to the very-ampleness of (220), (221), we can already obtain the very-amplenessof O P (T V | X ) ( κ ) for all large integers κ (cid:62) κ , for some non-e ff ective κ . In other words, the restrictedcotangent bundle Sym κ Ω V (cid:12)(cid:12)(cid:12) X is very ample on X for every κ (cid:62) κ . But to reach an explicit κ , onemay ask the Questions. (i)
Find one explicit (cid:96) ( N ). (ii) Find one explicit κ . Answer of (i).
The value (cid:96) ( N ) = z k z (cid:96) − j d (cid:16) z i z j (cid:17) ( i , j , k = ··· N , i (cid:44) j ) (222)guarantee the very-ampleness of O P (T P N K ) (1) ⊗ π ∗ O P N K ( (cid:96) ). Answer of (ii).
The second fact (B) above leads us to consider the semigroup G of the usualAbelian group Z ⊕ Z generated by elements ( (cid:96) , (cid:96) ) such that O P (T V | X ) ( (cid:96) ) ⊗ π ∗ O P N K ( (cid:96) ) is very mple. Then, the following elements are contained in G , for all m (cid:62) + (cid:16) n + n (cid:17) : [see (220) , ∀ b (cid:62) , a (cid:62) b + ] (cid:16) m a − n − r , − m b − N + + c (cid:88) i = d i + c + r (cid:88) i = d i (cid:17) , [see (221) , (cid:96) (cid:62) (cid:96) ( N ) = ] (1 , (cid:96) ) . Also, the first fact (A) above says that if ( (cid:96) , (cid:96) ) ∈ G , then ( (cid:96) , (cid:96) ) ∈ G for all (cid:96) (cid:62) (cid:96) . Thus,Question (ii) becomes to find one explicit κ such that ( κ, ∈ G for all κ (cid:62) κ .Paying no attention to optimality, taking: b = , a = , m = − N + + c (cid:88) i = d i + c + r (cid:88) i = d i + , we receive that ( m a − n − r , − ∈ G . Adding (1 , ∈ G , we receive ( m a − n − r + , ∈ G .Now, also using ( m a − n − r , ∈ G , recalling Observation 5.4, we may take: κ = ( m a − n − r −
1) ( m a − n − r ) (cid:54) a m , or the larger neater lower bound: κ = (cid:16) c (cid:88) i = d i + c + r (cid:88) i = d i (cid:17) . Thus, we have proved the Very-Ampleness Theorem 1.4 for K = C . Remembering that very-ampleness (or not) is preserved under any base change obtained by ambient field extension , andnoting the field extensions Q (cid:44) → C and Q (cid:44) → K for any field K with characteristic zero, bysome standard arguments in algebraic geometry, we conclude the proof of the Very-AmplenessTheorem 1.4.When K has positive characteristic, by the same arguments, we could also receive the samevery-ampleness theorem provided the similar results about the Fujita Conjecture hold over thefield K . R eferences [1] Benoist , O.:
Degrés d’homogénéité de l’ensemble des intersections complètes singulières , Ann. Inst. Fourier (Grenoble) (2012), no. 3, 1189–1214. [2] Berczi , G.:
Thom polynomials and the Green-Gri ffi ths conjecture , arxiv.org/abs/1011.4710 . [3] Bogomolov , F.:
Holomorphic symmetric tensors on projective surfaces , Uspekhi Mat. Nauk (1978), no. 5 (203), 171–172. [4] Bogomolov , F., de Oliveira , B.:
Symmetric tensors and geometry of P N subvarieties , Geom. Funct. Anal. (2008), no. 3,637–656. [5] Brotbek , D.:
Variétés projectives à fibré cotangent ample , Thèse de l’Université de Rennes 1, octobre 2011, 114 pp. [6]
Brotbek , D.:
Hyperbolicity related problems for complete intersection varieties , Compos. Math. (2014), no. 3, 369–395. [7]
Brotbek , D.:
Explicit symmetric di ff erential forms on complete intersection varieties and applications , arxiv.org/abs/1406.7848 . [8] Brückmann , P.,
Rackwitz , H.-G.:
T-symmetrical tensor forms on complete intersections , Math. Ann. (1990), no. 4,627–635. [9]
Brückmann , P.,
Rackwitz , H.-G.:
On cohomology of complete intersections with coe ffi cients in the twisted sheaf of di ff er-ential forms in the case of prime characteristic . Recent research on pure and applied algebra, 117–152, Nova Sci. Publ.,Hauppauge, NY, 2003. [10] Brunebarbe , Y.,
Klingler , B.,
Totaro , B.:
Symmetric di ff erentials and the fundamental group , Duke Math. J., (14),2797–2813, 2013. 92 Chen , H.-Y:
Positivité en géométrie algébrique et en géométrie d’Arakelov , Thèse de doctorat, École Polytechnique, Décem-bre 2006, 209 pp. [12]
Clemens , H.:
Curves on generic hypersurfaces , Annales Scientifiques de l’École Normale Supérieure. Quatrième Série (1986), no. 4, 629–636. [13] Conduché , D.,
Palmieri , E.:
On Chern ratios for surfaces with ample cotangent bundle , Matematiche (Catania) (2006),no. 1, 143–156. [14] Cox , D.,
Little , J.,
O’Shea , D.:
Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry andCommutative Algebra . Third edition, Undergraduate Texts in Mathematics, vol. . Springer, New York, 2007. xvi + [15] Darondeau , L.:
Fiber integration on the Demailly tower , arxiv.org/abs/1312.0109 , to appear in Annales de l’Institut Fourier. [16] Darondeau , L.:
On the Logarithmic Green-Gri ffi ths Conjecture , arxiv.org/abs/1402.1396 , to appear in International Mathe-matics Research Notices. [17] Darondeau , L.:
Slanted Vector Fields for Jet Spaces , arxiv.org/abs/1404.0212 , to appear in Mathematische Zeitschrift. [18] Debarre , O.:
Varieties with ample cotangent bundle , Compos. Math. (2005), no. 6, 1445–1459. [19]
Demailly , J.-P.:
A numerical criterion for very ample line bundles , J. Di ff . Geom. (1993), 323–374. [20] Demailly , J.-P.:
Algebraic criteria for Kobayashi hyperbolic projective varieties and jet di ff erentials , Algebraic geometry–Santa Cruz 1995, 285- 360, Proc. Sympos. Pure Math. , Part 2, Amer. Math. Soc., Providence, RI, 1997. [21] Demailly , J.-P.:
Méthodes L et résultats e ff ectifs en gémétrie algébrique , Séminaire Bourbaki, novembre 1998. [22] Demailly , J.-P.:
Holomorphic Morse inequalities and the Green-Gri ffi ths-Lang conjecture , Pure and Applied MathematicsQuarterly (2011), 1165–1208. [23] Demailly , J.-P.,
El Goul , J.:
Hyperbolicity of generic surfaces of high degree in projective 3-space , Amer. J. Math. (2000), 515–546 [24]
Diverio , S.: Di ff erential equations on complex projective hypersurfaces of low dimension , Compos. Math. (2008), no. 4,920–932. [25] Diverio , S.:
Existence of global invariant jet di ff erentials on projective hypersurfaces of high degree , Math. Ann. (2009),no. 2, 293–315. [26] Diverio , S.,
Merker , J.,
Rousseau , E.: E ff ective algebraic degeneracy , Invent. Math. (2010), no. 1, 161-223. [27] Diverio , S.,
Rousseau , E.:
A Survey on Hyperbolicity of Projective Hypersurfaces . Publicaçñes Matemáticas do IMPA.Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011. x +
109 pp. [28]
Diverio , S.,
Rousseau , E.:
The exceptional set and the Green-Gri ffi ths locus do not always coincide , arxiv.org/abs/1302.4756 . [29] Diverio , S.,
Trapani , S.:
A remark on the codimension of the Green-Gri ffi ths locus of generic projective hypersurfaces ofhigh degree , J. Reine Angew. Math. , 55–61, 2010. [30] Ein , L.:
Subvarieties of generic complete intersections , Invent. Math. (1988), no. 1, 163–169. [31] Eisenbud , D.:
Commutative algebra: with a view toward algebraic geometry . Graduate Texts in Mathematics, vol. .Springer-Verlag, New York, 1995. xvi +
785 pp. [32]
Eisenbud , D.,
Harris , J.:
The geometry of schemes . Graduate Texts in Mathematics, vol. . Springer-Verlag, New York,2000. x +
294 pp. [33]
Grothendieck , A.,
Dieudonné , J. A:
Éléments de géométrie algébrique. III. Étude Cohomologique des Faisceaux Cohérents.I . Publ. Math. IHÉS, no. , 1961. [34] Fujita , T.:
On polarized manifolds whose adjoint bundles are not semipositive , Algebraic geometry, Sendai, 1985, Adv. Stud.Pure Math. , North-Holland, Amsterdam, 167–178. [35] Harris , J.:
Algebraic geometry: a first course . Graduate Texts in Mathematics, vol. . Springer-Verlag, New York, 1992.xix +
328 pp. [36]
Hartshorne , R.:
Ample vector bundles , Inst. Hautes Études Sci. Publ. Math. (1966), 63–94. [37] Hartshorne , R.:
Algebraic geometry . Graduate Texts in Mathematics, vol. . Springer-Verlag, New York-Heidelberg, 1977.xvi +
496 pp. [38]
Lang , S.:
Hyperbolic and Diophantine analysis , Bull. Amer. Math. Soc. (N.S.) (1986), no. 2, 159–205. [39] Lang , S.:
Algebra . Revised third edition. Graduate Texts in Mathematics, vol. . Springer-Verlag, New York, 2002.xvi +
914 pp. [40]
Lazarsfeld , R.:
Positivity in algebraic geometry. I. Classical setting: line bundles and linear series . Ergebnisse der Mathe-matik und ihrer Grenzgebiete, vol. . Springer-Verlag, Berlin, 2004. xviii +
387 pp.93
Lazarsfeld , R.:
Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals . Ergebnisse derMathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. . Springer-Verlag, Berlin,2004. xviii +
385 pp. [42]
Liu , Q.:
Algebraic geometry and arithmetic curves . Oxford Graduate Texts in Mathematics, vol. . Oxford Science Publica-tions. Oxford University Press, Oxford, 2002. xvi +
576 pp. [43]
Merker , J.:
Low pole order frames on vertical jets of the universal hypersurface , Ann. Inst. Fourier (Grenoble) (2009),no. 3, 1077–1104. [44] Merker , J.:
Siu-Yeung jet di ff erentials on complete intersection surfaces X in P ( C ), arxiv.org/abs/1312.5688 . [45] Merker , J.:
Extrinsic projective curves X in P ( C ) : harmony with intrinsic cohomology , arxiv.org/abs/1402.1108 . [46] Merker , J.:
Rationality in di ff erential algebraic geometry , Proceedings of the Abel Symposium 2013, John-Erik Fornæssand Erlend Wold Eds., Springer-Verlag, 2015, to appear, 47 pages, arxiv.org/abs/1405.7625 . [47] Miyaoka , Y.:
Algebraic surfaces with positive indices , Classification of algebraic and analytic manifolds (Katata, 1982),281–301, Progr. Math., vol. , Birkhäuser Boston, Boston, MA, 1983. [48] Moriwaki , A.:
Remarks on rational points of varieties whose cotangent bundles are generated by global sections , Math. Res.Lett. (1995), no. 1, 113–118. [49] Moriwaki , A.:
Geometric height inequality on varieties with ample cotangent bundles , J. Algebraic Geom. (1995), no. 2,385–396. [50] Mourougane , C.:
Families of hypersurfaces of large degree , J. Eur. Math. Soc. (JEMS) (2012), no. 3, 911–936. [51] Noguchi , J.,
Sunada , T.:
Finiteness of the family of rational and meromorphic mappings into algebraic varieties , Amer. J.Math (1982), 887-990. [52]
P˘aun , M.:
Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity , Mathematische Annalen (2008), 875–892. [53]
P˘aun , M.:
Techniques de construction de di ff érentielles holomorphes et hyperbolicité [d’après J.-P. Demailly, S. Diverio, J.Merker, E. Rousseau, Y.-T. Siu...] , Séminaire Bourbaki, Octobre 2012, no. 1061, 35 pp. [54] Peskine , C.:
An algebraic introduction to complex projective geometry. 1: commutative algebra . Cambridge Studies inAdvanced Mathematics, vol. . Cambridge University Press, Cambridge, 1996. x +
230 pp. [55]
Roulleau , X.,
Rousseau , E.:
Canonical surfaces with big cotangent bundle , Duke Math. J., (7), 1337–1351, 2014. [56]
Rousseau , E.:
Weak analytic hyperbolicity of generic hypersurfaces of high degree in P , Ann. Fac. Sci. Toulouse Math. (6) (2007), no. 2, 369–383. [57] Sakai , F.:
Symmetric powers of the cotangent bundle and classification of algebraic varieties , Algebraic geometry (Proc.Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol , 545–563, Springer, Berlin. [58]
Schneider , M.:
Symmetric di ff erential forms as embedding obstructions and vanishing theorems , J. Algebraic Geom. (1992), no. 2, 175–181. [59] Shafarevich , I. R:
Basic algebraic geometry. 1. Varieties in projective space . Second edition. Translated from the 1988Russian edition and with notes by Miles Reid. Springer-Verlag, Berlin, 1994. xx +
303 pp. [60]
Siu , Y.-T.:
Hyperbolicity problems in function theory, Five decades as a mathematician and educator , 409–513, World Sci.Publ., River Edge, NJ, 1995. [61]
Siu , Y.-T.: E ff ective very ampleness , Invent. Math. (1996), 563–571. [62] Siu , Y.-T.:
Some recent transcendental techniques in algebraic and complex geometry , Proceedings of the InternationalCongress of Mathematicians, Vol. I (Beijing, 2002), 439–448, Higher Ed. Press, Beijing, 2002. [63]
Siu , Y.-T.:
Hyperbolicity in complex geometry , The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 543–566. [64]
Siu , Y.-T.:
Hyperbolicity of generic high-degree hypersurfaces in complex projective space , Invent. Math., pp. 1–98, 2015. [65]
Siu , Y.-T.,
Yeung , S.-K:
Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projectiveplane , Invent. Math. (1996), 573–618. [66]
Voisin , C.:
On a conjecture of Clemens on rational curves on hypersurfaces , J. Di ff . Geom. (1996), no. 1, 200–213. L aboratoire de M ath ´ ematiques d ’O rsay , U niversit ´ e P aris -S ud (F rance ) E-mail address : [email protected]@math.u-psud.fr