Analysis on some infinite modules, inner projection, and applications
aa r X i v : . [ m a t h . AG ] M a y ANALYSIS ON SOME INFINITE MODULES, INNERPROJECTION, AND APPLICATIONS
KANGJIN HAN AND SIJONG KWAK
Dedicated to the memory of Hyo Chul Myung(June 15, 1937 - February 11, 2010)
Abstract.
A projective scheme X is called ‘quadratic’ if X is scheme-theoreticallycut out by homogeneous equations of degree 2. Furthermore, we say X satisfies‘property N ,p ’ if it is quadratic and the quadratic ideal has only linear syzygiesup to first p -th steps. In the present paper, we compare the linear syzygies ofthe inner projections with those of X and obtain a theorem on ‘embedded linearsyzygies’ as one of our main results. This is the natural projection-analogue of‘restricting linear syzygies’ in the linear section case, [EGHP05]. As an immediatecorollary, we show that the inner projections of X satisfy property N ,p − forany reduced scheme X with property N ,p .Moreover, we also obtain the neccessary lower bound (codim X ) · p − p ( p − , which is sharp, on the number of quadrics vanishing on X in order to satisfy N ,p and show that the arithmetic depths of inner projections are equal to thatof the quadratic scheme X . These results admit an interesting ‘syzygetic’ rigiditytheorem on property N ,p which leads the classifications of extremal and next toextremal cases.For these results we develope the elimination mapping cone theorem for infin-itely generated graded modules and improve the partial elimination ideal theoryinitiated by M. Green. This new method allows us to treat a wider class of pro-jective schemes which can not be covered by the Koszul cohomology techniques,because these are not projectively normal in general. Keywords: linear syzygies, the mapping cone theorem, partial eliminationideals, inner projection, arithmetic depth, Castelnuovo-Mumford regularity.
Contents
Introduction 21. Definitions and Preliminaries 32. Elimination mapping cone construction and Partial elimination ideals 53. Embedded linear syzygies and Applications 104. Arithmetic depth and Syzygetic rigidity 145. Examples and Open questions 18References 21
Mathematics Subject Classification.
Primary 14N05, 13D02, 14N25; Secondary 51N35.The authors were supported by Basic Science Research Program through the National ResearchFoundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grantNo.2009-0063180).
Introduction
Let X be a nondegenerate reduced closed subscheme in a projective space P N over an algebraically closed field k of characteristic zero and R = k [ x , . . . , x N ]be the coordinate ring of P N . The equations defining X and the syzygies amongthem have played a central role to study projective schemes in algebraic geometry.Further the syzygy structures and their geometric implications have been intensivelyfocused for the most interesting case, i.e. projective schemes having property N ,p for last twenty years, see [CKK06, EGHP05, EGHP06, EHU06, GL88]. They areclosely related to the Eisenbud-Goto conjecture on Castelnuovo-Mumford regularityand other conjectures on linear syzygies in classical algebraic geometry. The linearsections and projections of X have been very useful to understand those problems.For the linear sections, we have interesting results on ‘Restricting linear syzygies’due to Eisenbud, Green, Hulek, Popescu, see [EGHP05]. Along this line, a naturalquestion could be raised: What is the relations between the syzygies of projections and X ? In the present paper, we especially consider the relations between the linearsyzygies of inner projections and those of X . Note that the inner projection hasbeen a standard issue classically since del Pezzo and Fano used this projection forthe classification of del Pezzo surfaces and Fano 3-folds, see [Reid00]. There are alsosome known results about non-birational loci of these projection morphisms andgeometric structures of the projection images, see [BHSS00, CC01, Seg36, Som79]. Problems
We list our main problems in detail:(a) (Embedded linear syzygies) Let X be a nondegenerate reduced scheme in P N satisfying property N ,p , ( p ≥ ⊂ P N of dim Λ = t < p with Λ ∩ X = ∅ , h Λ ∩ X i = Λ and X Λ = π Λ ( X \ Λ) in P N − t − . How do the syzygies behave under projections?D. Eisenbud et. al. showed that under some N ,p -assumption, the syzygiesof X restrict surjectively to the syzygies of linear sections in their paper‘Restricting linear syzygies’, [EGHP05]. Is there any natural projection-analogue of the linear section case? Bearing on this problem, we also expectthat X Λ satisfies property N ,p − t − .(b) (Necessary lower bound for property N ,p ) For a quadratic scheme X satis-fying N ,p , it is roughly believed that the more quadratic equations X has,the further steps linear syzygies proceed to. Therefore one can ask ‘howmany quadrics does X require to satisfy property N ,p ?’ This is a naturalquestion, but not yet known.(c) (Syzygetic rigidity theorem) In [EGHP05, EGHP06] they also show that aclosed subscheme X ⊂ P N is 2-regular if X satisfies property N , codim X andcharacterize all 2-regular algebraic sets geometrically for this extremal case.What about ‘next to extremal case’, i.e. a scheme X satisfying N , codim X − ?How to classify or characterize them in a suitable category?For those problems, we develope the elimination mapping cone theorem for in-finitely generated graded modules and improve the partial elimination ideal theoryinitiated by M. Green for the inner projection. This allows us to treat a widerclass of projective schemes which can not be covered by the Koszul cohomologytechniques, because these are not projectively normal in general. We have also NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 3 found it very interesting to understand some relations between the syzygies of itsprojections and those of X as we move the center of projection. Organization of the paper
We recall basic definitions and preliminaries inSection 1. In Section 2, we set up the elimination mapping cone construction for in-finitely generated graded modules and the partial elimination ideal theory for innerprojection which are crucial to understand the syzygy structures of infinitely gener-ated graded modules. This partial elimination ideal theory gives us local informationof X near the center of projection q ∈ X which turns out to govern syzygies andother properties of the inner projection X q from the ( global ) homogeneous equations.In Section 3, we obtain some results on syzygy structures and geometric prop-erties of inner projections, i.e. embedded linear syzygies, the number of quadraticequations, and their corollaries. In particular, we can show that for any projectivereduced scheme X satisfying property N ,p the inner projection from any smoothpoint satisfies at least property N ,p − and X Λ satisfies at least N ,p − t − for a gen-eral t -dimensional linear subspace Λ with dim X ∩ Λ = 0 (see Corollary 3.4 andRemark 3.5). We also give the neccessary lower bound (codim X ) · p − p ( p − on thenumber of quadrics vanishing on X in order to satisfy property N ,p , which is sharp.In Section 4 we prove that the arithmetic depths of inner projections are equal tothat of the given quadratic scheme. Combined with results in the previous section,this depth theorem leads us to a very interesting ‘syzygetic’ rigidity theorem on prop-erty N ,p in the category of varieties, namely, for the extremal (i.e. p = codim X )and next to extremal (i.e. p = codim X −
1) cases those varieties should be arith-metically Cohen-Macaulay (abbr. ACM) and we can give the classfications of thetwo cases. We also extend this result to more general category (See Corollary 4.5and Question 5.6). Finally, in Section 5 we see some examples and open questionsstimulating further work.
Acknowledgements
The first author would like to thank Professor Frank-OlafSchreyer for hosting his visit to Saarbr¨ucken under KOSEF-DAAD Summer InstituteProgram and for many valuable comments preparing this paper. The second authorwould like to thank Professor B. Sturmfels, M. Brodmann for their useful comments,especially P. Schenzel for valuable discussion and Example 5.1 during their stay inKorea Institute of Advanced Study(KIAS) and KAIST, Korea in the Summer 2009.We would also like to thank Professor F. Zak who informed us of Professors A. Alzatiand J.C. Sierra’s recent paper [AS10] related to our paper (see Remark 3.10).1.
Definitions and Preliminaries
We work over an algebraically closed field k of characteristic zero. Let X be anondegenerate reduced closed subscheme in a projective space P N . Definition 1.1.
Let X be as above.(a) X is said to be a quadratic scheme if there is a homogeneous ideal I gener-ated by equations of degree 2 which defines X scheme-theoretically (i.e. itssheafification e I is equal to the ideal sheaf I X of X ).(b) X is said to satisfy property N ,p scheme-theoretically if it is quadratic andthe quadratic ideal I has only linear syzygies at least up to first p -th steps. K. HAN AND S. KWAK (c) X is said to be m -regular if H i ( I X ( m − i )) = 0 for all i ≥
1. We callreg( X ) := min { m | H i ( I X ( m − i )) = 0 for all i ≥ } Castelnuovo-Mumfordregularity of X .Note that Definition 1.1 (b) is a generalization of known notions. It is the sameas property N p defined by Green-Lazarsfeld if X is projectively normal and I issaturated (see [GL88]) .Let π Λ : X ⊂ P N P N − t − denote the projection of X from a linear spaceΛ = P t . We call it either outer projection if X ∩ Λ = ∅ or inner projection in caseΛ ⊂ X . Every projection π Λ can be regarded as succesive compositions of suitableouter and inner projections from points. These projections as well as blow-ups havebeen very useful projective techniques in algebraic geometry. We briefly review thepreliminaries about an inner projection from a point q ∈ X .Let σ : e X → X be a blowing up of X at a smooth point q ∈ X . One has theregular morphism π ′ : e X ։ X q := π q ( X \ { q } ) ⊂ P N − with the following diagram; P N × P N − ⊃ e X π ′ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ σ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ P N ⊃ X π q / / ❴❴❴❴❴❴❴❴ X q = π q ( X \ q ) ⊂ P N − Classically, one says that a smooth variety X admits an inner projection if π ′ is an embedding for some point q ∈ X . This is equivalent to q ∈ X \ Trisec( X )where Trisec( X ) is the union of all lines ℓ with the condition that ℓ ⊂ X or X ∩ ℓ is a subscheme of length at least 3. We also know that the exceptional divisor E islinearly embedded via π ′ in P N − (i.e. π ′ ( E ) = P r − ⊂ P N − , r = dim X ) for anysubvariety X if the center q is smooth (see [Bau95, FCV99]).Let R = k [ x , . . . , x N ] and S = k [ x , x . . . , x N ] be the homogeneous coordinaterings of P N and P N − . Assume q = (1 : 0 : . . . : 0) ∈ X (by suitable coordinatechange). Let I be an ideal of R defining a reduced scheme X scheme-theoretically.Naturally we can give a scheme structure on the image X q by the ideal J := I T S .Note that the ideal J is reduced if I is reduced.In case of inner projection we note that R/I is not finitely generated S -module,because q ∈ X and there is no polynomial of the form f = x n + ( other terms ) forsome n ∈ N in the ideal I , even though R/I is finitely generated as R -module. I isalso an infinitely generated graded S -module with the following resolution : · · · → L ∞ j =2 S ( − i − j ) β i,j → · · · → L ∞ j =2 S ( − j ) β ,j → I → . In Section 2, we show that they have interesting syzygy structures as S -modules(see Proposition 2.5 and Remark 2.6).On the other hand, if X is quadratic, then we can write the quadratic ideal I as(1.1) I = ( x ℓ − Q , , . . . , x ℓ t − Q ,t , Q , . . . , Q s ) , q = (1 , , . . . , ∈ X NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 5 where ℓ i is a linear form, Q ,i , Q j are quadratic forms in S = k [ x , . . . , x N ] andthey are minimal generators. We can also assume all { ℓ i } are linearly inde-pendent, and all { Q ,i } are distinct. Clearly, { ℓ i } generate ( T q X ) ∗ . Note that t = codim( X ) = N − dim X if q is a smooth point. In general, t is equal to N − dim T q X . Convention
We are working on the following convention: • Let X ⊂ P N and q ∈ X be as above and I be a homogeneous defining ideal of X . We denote the S -ideal I ∩ S by J which gives the natural induced schemestructure on the projection image X q ⊂ P N − and call J the x -eliminationideal of I . In addition, we write the saturated ideal defining X (resp. X q )as I X (resp. I X q ). • (Betti numbers) For a graded R -module M , we define graded Betti numbers β Ri,j ( M ) of M by dim k Tor Ri ( M, k ) i + j . We consider β Si,j ( N ) for any graded S -module N in the same manner. We remind readers that Tor Ri ( R/I, k ) i + j =Tor Ri − ( I, k ) i − j +1 . So β Ri,j ( R/I ) = β Ri − ,j +1 ( I ). We write β Ri,j ( M ) as β i,j ( M ) or β i,j if it is obvious. • We often abbreviate Tor Ri ( M, k ) i + j as Tor Ri ( M ) i + j (same for S -module Tor). • (Arithmetic depth) When we refer the depth of X , denoted by depth R ( X ),we mean the arithmetic depth of X , i.e. depth R ( R/I X ).2. Elimination mapping cone construction and Partial eliminationideals
In general the mapping cone of the chain map between two complexes is a kind ofnatural extension of complexes induced by the given chain map. Now we constructsome graded mapping cone which we call ‘Elimination mapping cone’. This isnaturally related to projections and very useful to understand the syzygies of pro-jections. Another ingredient is the partial elimination ideal theory. Let us constructthe graded mapping cone theorem and consider the partial elimination ideal theoryfrom a viewpoint of inner projections.
Elimination Mapping Cone Construction . Let W = k h x , · · · , x N i ⊂ V = k h x , · · · , x N i be vector spaces over k and S = Sym( W ) = k [ x , . . . , x N ] ⊂ R =Sym( V ) = k [ x , . . . , x N ] be polynomial rings. • M : a graded R -module given a degree 1 shifting map by µ (i.e. µ : M i → M i +1 ) • G ∗ (resp. F ∗ ) : the graded Koszul complex of M , K S ∗ ( M )(resp. M [ − K S ∗ ( M [ − → ∧ N W ⊗ M → · · · → ∧ W ⊗ M → W ⊗ M → M → G i ) i + j are K Si ( M ) i + j = ∧ i W ⊗ M j (resp. ( F i ) i + j = ∧ i W ⊗ M j − ). • Then µ : M i → M i +1 induces the chain map¯ µ : F ∗ = K S ∗ ( M [ − −→ G ∗ = K S ∗ ( M ) of degree 0.Now we construct the mapping cone ( C ¯ µ , d ¯ µ ) such that(2.1) 0 −→ G ∗ −→ ( C ¯ µ ) ∗ −→ F ∗ [ − −→ , K. HAN AND S. KWAK where C ¯ µ is a direct sum G ∗ L F ∗ [ −
1] and the differential d ¯ µ is given by( d ¯ µ ) ∗ = (cid:18) ∂ G ( − ∗ +1 ¯ µ ∂ F (cid:19) , where ∂ is the differential of Koszul complex. From the construction, it can bechecked that we have the following isomorphism (see [AK11]):Tor Ri ( M ) i + j ≃ H i (( C ¯ µ ) ∗ ) i + j . Suppose M is a graded R -module which is also a graded S -module. Consider amultiplication map µ = × x as a naturally given degree 1 shifting map on M . In thiscase, the long exact sequence on homology groups induced from (2.1) is importantand very useful to study the syzygies of projections. Note that in general we candefine property N d,p similarly (i.e. β Ri,j ( R/I ) = 0 for any ≤ i ≤ p, j ≥ d ). Theorem 2.1. (Elimination mapping cone sequence)
Let S = k [ x , . . . , x N ] ⊂ R = k [ x , x . . . , x N ] be two polynomial rings. (a) Let M be a graded R -module which is not necessarily finitely generated.Then, we have a natural long exact sequence: · · · Tor Ri ( M ) i + j → Tor Si − ( M ) i − j ¯ µ → Tor Si − ( M ) i − j +1 → Tor Ri − ( M ) i − j +1 · · · whose connecting homomorphism ¯ µ is induced by the multiplicative map × x . (b) Assume that
R/I satisfies property N d,p for some d ≥ , p ≥ . Then amultiplication by x induces a sequence of isomorphisms on Tor Si ( I ) i + j for ≤ i ≤ p − , j ≥ d + 1 and a surjection for j = d ; · · · × x → Tor Si ( I ) i + d × x ։ Tor Si ( I ) i + d +1 × x ∼ → Tor Si ( I ) i + d +2 × x ∼ → · · · . For i = p − , we have a sequence of surjections from j = d · · · × x → Tor Sp − ( I ) p − d × x ։ Tor Sp − ( I ) p − d +1 × x ։ Tor Sp − ( I ) p − d +2 × x ։ · · · Remark 2.2.
J. Ahn and the second author pointed out that this graded mappingcone construction is closely related to outer projections (see [AK11]). We remarkhere that this theorem is also true even for an infinitely generated S -module M and relates the torsion module Tor R ( M ) to the torsion module of M as S -module.Therefore this gives us useful information about syzygies of inner projections. Proof. (a) follows from theorem 2.2 in [AK11]. For a proof of (b), consider themapping cone sequence of Theorem (2.1) for M = I Tor Ri +1 ( I ) i +1+ j → Tor Si ( I ) i + j × x −→ Tor Si ( I ) i + j +1 → Tor Ri ( I ) i + j +1 Note that Tor Ri ( I, k ) i + j = 0 for 0 ≤ i ≤ p − j ≥ d + 1 by assumption that I has N d,p property as a R -module. So, we have an isomorphismTor Si ( I, k ) i + j × x ∼ → Tor Si ( I, k ) i + j +1 for 0 ≤ i ≤ p − ∀ j ≥ d + 1 and a surjection for j = d .In case i = p −
1, we know Tor Rp − ( I ) p − j = 0 for j ≥ d + 1 in the mapping conesequenceTor Rp ( I ) p + j → Tor Sp − ( I ) p − j × x −→ Tor Sp − ( I ) p − j +1 → Tor Rp − ( I ) p − j +1 . Therefore we get the desired surjections for i = p − (cid:3) NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 7
Partial Elimination Ideals under a Projection . Mark Green introduced partialelimination ideals in his lecture note [Gre98]. For the degree lexicographic order, if f ∈ I m has leading term in( f ) = x d · · · x d n n , we set d ( f ) = d , the leading powerof x in f . Then we can give the definition of partial elimination ideals as in thefollowing. Definition 2.3.
Let I ⊂ R be a homogeneous ideal and let e K i ( I ) = M m ≥ (cid:8) f ∈ I m | d ( f ) ≤ i (cid:9) . If f ∈ e K i ( I ), we may write uniquely f = x i ¯ f + g where d ( g ) < i . Now we definethe ideal K i ( I ) in S generated by the image of e K i ( I ) under the map f ¯ f and wecall K i ( I ) the i -th partial elimination ideal of I . Observation 2.4.
We can observe some properties of these ideals in the projectioncase.(a) 0–th partial elimination ideal K ( I ) of I is J := I ∩ S = M m ≥ (cid:8) f ∈ I m | d ( f ) = 0 (cid:9) . Note that the ideal J gives a scheme structure on the image X q naturally.(b) e K i ( I ) is a natural filtration of I with respect to x which also induces afiltraton on K i ( I )’s : J = e K ( I ) ⊂ e K ( I ) ⊂ · · · ⊂ e K i ( I ) ⊂ · · · ⊂ e K ∞ ( I ) = IJ = K ( I ) ⊂ K ( I ) ⊂ · · · ⊂ K i ( I ) ⊂ · · · ⊂ S. (c) e K i ( I ) is a finitely generated graded S -module and there is a short exactsequence as graded S -modules(2.2) 0 → e K i − ( I ) e K ( I ) → e K i ( I ) e K ( I ) → K i ( I )( − i ) → . In general we can see at least when the K i ( I )’s stabilize and what they look likefor inner projections. The following proposition is the anwser. It also tells us aminimal free resoultion for some infinitely generated graded S -module which is veryuseful to understand the defining equations and syzygies of inner projections. Proposition 2.5.
Let X ⊂ P N be a reduced projective scheme with a homogeneousdefining ideal I . Let q = (1 , , . . . , ∈ X . (a) If I satisfies property N d, , K i ( I ) stabilizes at least at i = d − to an idealdefining T C q X , the tangent cone of X at q . So if q is smooth, K d − ( I ) consists of linear forms which defines T q X . (b) In particular, if I is generated by quadrics and q is smooth, then K i ( I ) stabilizes at i = 1 step to an ideal I L = ( l , . . . , l e ) , e = codim( X, P N ) whichdefines the tangent space T q X , i.e. J = K ( I ) ⊂ I L = K ( I ) = · · · = K i ( I ) = · · · ⊂ S and I/J has obvious syzygies as an infinitely generated
K. HAN AND S. KWAK S -module such that: S ( − e − b e S ( − b S ( − b → ⊕ S ( − e − b e → · · · → ⊕ S ( − b → ⊕ S ( − b → I/J → , ⊕ S ( − e − b e ⊕ S ( − b ⊕ S ( − b · · · · · · · · · where b i = (cid:0) ei (cid:1) .Proof. (a) Since I is generated in deg ≤ d and q = (1 , , . . . , ∈ X , we havegenerators { F i } of I with d ( F i ) ≤ d − x d + other lower terms in x . From this, every leading term f of a homogeneouspolynomial F in I of deg k ( k ≥ d ) is written as x (cid:3) · ¯ f where ¯ f ∈ K c ( I ) for some c ≤ d −
1. So K i ( I ) stabilizes at least at i = d −
1. Note that all ¯ f ∈ K i ( I ) ( i ≥ T C q X of X at q becausethey come from f = x i ¯ f + g ∈ I , d ( g ) < i . Therefore, K i ( I ) stabilizes to the idealdefining T C q X . In case of a smooth point q ∈ X , T q X = T C q X and K d − ( I ) =( ℓ , . . . , ℓ e ), e = codim( X, P N ).(b) Since d = 2 and q is a smooth point, K i ( I ) becomes I L = ( ℓ , . . . , ℓ e ) for each i ≥
1. For the sake of the S -module syzygy of I/J , first note that I = e K ∞ ( I ). Fromthe exact sequence (2.2), we get e K ( I ) /J ≃ K ( I )( −
1) with the following linearKoszul resolution: letting b i = (cid:0) ei (cid:1) ,0 → S ( − e − b e → · · · → S ( − b → S ( − b → e K ( I ) /J → . Next, K ( I )( −
2) = K ( I )( −
2) has also linear syzygies:0 → S ( − e − b e → · · · → S ( − b → S ( − b → K ( I )( − → , and we have the following exact sequence from (2.2) again,0 → e K ( I ) J → e K ( I ) J → K ( I )( − → . By the long exact sequence of Tor, we know that S ( − e − b e S ( − b S ( − b → ⊕ S ( − e − b e → · · · → ⊕ S ( − b → ⊕ S ( − b → ˜ K ( I ) /J → . Similarly, we can compute the syzygy of e K i ( I ) /J for any i , and we get the desiredresolution of I/J = e K ∞ ( I ) /J as S -module in the end. (cid:3) Remark 2.6.
For the next section, we remark some useful facts as follows:(a) (Reduction of syzygies)
From the sequence (2.2), we have an isomorphism(2.3) Tor Si ( I/J ) i + j ≃ Tor Si ( e K d ( I ) /J ) i + j for any d ≥ j − . In other word, the syzygies of an infinitely generated S -module I/J can becomputed from the syzygies of finitely generated S -module e K i ( I ). Further,if all K i ( I ) ’s allow only linear syzygies at each step (i.e. Tor Sa ( K i ( I )) a + b = 0for ∀ a and ∀ b = i + 1), thenTor Si ( I/J ) i + j ≃ Tor Si ( K j − ( I )( − j + 1)) i + j for any i, j as Proposition 2.5 (b) shows us that the syzygies of I/J essentially comejust from the Koszul syzygies of { x α ℓ , . . . , x α ℓ e } of K α ( I ). NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 9 (b) (Commutativity of x -multiplication) Consider the S -module homo-morphism φ : I → I/J , the natural quotient map and also consider multipli-cation maps in both I and I/J . This multiplication × x is not well-definedin I/J , while it is a well-defined S -module homomorphism in I . But if X isquadratic and q is a smooth point, then, by Proposition 2.5 (b) and above(a), we have a commuting diagram in Tor-level:Tor Si ( I ) i + j +1 φ / / Tor Si ( I/J ) i + j +1 ψisom. / / Tor Si ( L j − ( I )) i + j +1 Tor Si ( I ) i + j φ / / × x O O Tor Si ( I/J ) i + j ∃ × x O O ✤✤✤ ψisom. / / Tor Si ( L j − ( I )) i + j , × x O O where L j − ( I ) := K j − ( I )( − j + 1) and each row ψ ◦ φ is induced from the S -homomorphism I → L j − ( I ) given by f = x j − ¯ f j − + x j − ¯ f j − + · · · + ¯ f x j − ¯ f j − which naturally commutes with the x -multiplication. Remark 2.7. (Outer projection case) (a) We can also consider outer projection by the similar method. In thiscase K i ( I ) always stabilizes at least at d − th step to (1) = S if I satis-fies N d, . More interesting fact is that K d − ( I ) consists of linear formswith N d, -condition. Especially, suppose that X satisfy property N , and q = (1 , , · · · , / ∈ X . Then K ( I ) is an ideal of linear forms I Σ defining thesingular locus Σ of π q in X q ⊂ P N − (see [AK11] for details). By the similarmethod as in the inner projection, we see that I/J has simple S -modulesyzygies such that:0 → S ( − t − b t → · · · → S ( − b → S ( − b +1 → I/J → , ⊕ S ( − ⊕ S ( − · · · where b i = (cid:0) ti (cid:1) , t = codim(Σ , P N − ). So, this resolution can be used to studythe outer projection case.(b) The stabilized ideal gives an important information for projections. In outercase of N ,p ( p ≥ K ( I ) also shows us the tangential behavior of X at q and T C q X plays an important role in our problem.Now there arise some basic and natural questions. How are the syzygies of J related to the S -module syzygies of I and to the R -module syzygies of I ? Withthe assumption for property N ,p , we may ask the following question specifically :Is J generated only by quadrics if so I is? There might be cubic generators like ℓ i Q ,j − ℓ j Q ,i (= ℓ j · [ x ℓ i − Q ,i ] − ℓ i · [ x ℓ j − Q ,j ] ) in J (see (1.1) in Section 1).If not, how about the case of N , ? What can we say about higher linear syzygiesof X q ? We will answer these kind of syzygy and elimination problems and derivestronger results by using the elimination mapping cone sequence and the partialelimination ideal theory in next section. Embedded linear syzygies and Applications
Recently, D. Eisenbud et. al. showed that with assumption for some property N ,p , the syzygies of X restrict surjectively to the syzygies of linear sections intheir paper ‘Restricting linear syzygies’, [EGHP05]. We consider in this section thebehavior of the syzygies under inner projections and we present one of our maintheorems on ‘embedded linear syzygies’ which is the natural projection-analogue ofthe linear section case. Theorem 3.1.
Let X ⊂ P N be a nondegenerate reduced quadratic scheme whosesaturated ideal I X satisfies property N ,p for some p ≥ and q ∈ X be a smoothpoint. Consider the inner projection π q : X X q ⊂ P N − . Then there is aninjection between the minimal free resolutions of I X q and I X up to first ( p − -thstep, i.e. ∃ f : Tor Si ( I X q , k ) i + j ֒ → Tor Ri ( I X , k ) i + j for ≤ i ≤ p − , ∀ j ∈ Z which is induced by the natural inclusion I X q ֒ → I X and the elimination mappingcone sequence (see Theorem 2.1 (a)). Remark 3.2.
The method used to prove Theorme 3.1 is, in fact, available for anyideal I defining X and J = I ∩ S defining X q scheme-theoretically. Proof.
We have a basic short exact sequence of S -modules,(3.1) 0 −→ I X q −→ I X −→ I X /I X q −→ . From the long exact sequence of (3.1) and the mapping cone sequence of I X in (2.1), we have a diagram0 (cid:15) (cid:15) (cid:15) (cid:15) Tor Si ( I X q , k ) i + j − (cid:15) (cid:15) Tor Si ( I X q , k ) i + jg (cid:15) (cid:15) f := h ◦ g ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ Tor Si ( I X , k ) i + j − (cid:15) (cid:15) × x / / Tor Si ( I X , k ) i + j (cid:15) (cid:15) h / / Tor Ri ( I X , k ) i + j Tor Si ( I X /I X q , k ) i + j − × x / / Tor Si ( I X /I X q , k ) i + j For any 0 ≤ i ≤ p −
2, we proceed with j case by case. Case 1) j ≤
1: Since Tor Si ( I X q , k ) i + j = 0, so it is obviously injected toTor Ri ( I X , k ) i + j by f . Case 2) j = 2 (i.e. linear syzygy cases for each i ): From (3.1), we haveTor Si +1 ( I X /I X q , k ) i +2 −→ Tor Si ( I X q , k ) i +2 g −→ Tor Si ( I X , k ) i +2NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 11 Since q is a smooth point, with N , condition we know the syzygy structures of I X /I X q as S -module by Proposition 2.5 (b). It shows Tor Si +1 ( I X /I X q , k ) i +2 = 0,implying that g is injective. Since X is nondegenerate, so Tor Si ( I X , k ) i +1 = 0 and h is also injective at the horizontal mapping cone sequence of above diagram. Hence f is injective in this case, too. Case 3) j ≥
3: First note that Tor Ri ( I X , k ) i + j = 0 for 0 ≤ i ≤ p − , j ≥ N ,p . We will show that g is injective and Tor Si ( I X , k ) i + j is isomorphic to Tor Si ( I X /I X q , k ) i + j for 0 ≤ i ≤ p −
2. Then we can conclude thatTor Si ( I X q , k ) i + j = 0, so f is injective for 0 ≤ i ≤ p − x sufficiently, we havethe following diagram:Tor Si +1 ( I X , k ) i +1+ j − ∴ surj. / / / / surj. (cid:15) (cid:15) (cid:15) (cid:15) Tor Si +1 ( I X /I X q , k ) i +1+ j − / / isom. (cid:15) (cid:15) Tor Si ( I X q , k ) i + j g / / Tor Si +1 ( I X , k ) i +1+ N isom. / / Tor Si +1 ( I X /I X q , k ) i +1+ N The left vertical map is surjective from Theorem 2.1 (b), and the right one isan isomorphism by the syzygy structures of I X /I X q in Proposition 2.5 (b). Since I X q is a finite S -module, Tor Si ( I X q , k ) i + N = 0 for sufficiently large N , so we get thebelow(second row) isomorphism. Therefore the mapTor Si +1 ( I X , k ) i +1+ j − → Tor Si +1 ( I X /I X q , k ) i +1+ j − is surjective, and g is injective.Similarly, we can have the desired isomorphism between Tor Si ( I X , k ) i + j andTor Si ( I X /I X q , k ) i + j as follows:Tor Si ( I X q , k ) i + j g / / Tor Si ( I X , k ) i + j αisom. / / isom. (cid:15) (cid:15) Tor Si ( I X /I X q , k ) i + jisom. (cid:15) (cid:15) Tor Si ( I X , k ) i + N isom. / / Tor Si ( I X /I X q , k ) i + N In this case, the mapping cone construction gives the left vertical isomorphismby Theorem 2.1 (b). So the above map α is an isomorphism as we wish, andTor Si ( I X q , k ) i + j = 0 for 0 ≤ i ≤ p − , j ≥ (cid:3) This main Theorem 3.1 tell us that all the S -module syzygies of X q are exactly thevery ones which are already embedded in the linear syzygies of X as R -module. Thisdoesn’t hold for outer projection and inner projection of varieties with N d,p ( d ≥ Example 3.3.
Let C be a rational normal curve in P and I C be the homogeneousideal ( x x − x , x x − x x − x , x − x x − x x ) under suitable coordinatechange. We know C is 2-regular and consider an outer projection of C from q =(1 , , , I C q = ( x − x x x − x ) has a cubic generator (i.e. N , ). Since x − x x x − x = ( − x ) · [ x x − x ] + x · [ x x − x x − x ] , this is zero inTor R ( I C ) and Tor S ( I C q ) → Tor R ( I C ) is not injective. In general, if we take thecenter q ∈ L ≃ P which is a multisecant ( e.g. at least -secant ) 2-plane, then forouter and inner projection cases there is a multisecant line to X q . So, the definingequations of X q may have larger degrees.As an immediate consequence, we have the following corollary. Corollary 3.4. (Property N ,p − of inner projections) Let X ⊂ P N be a non-degenerate reduced quadratic scheme satisfying property N ,p for some p ≥ and q ∈ X be a smooth point. Then, the inner projection X q is also quadratic andsatisfies property N ,p − .Proof. Case 3) j ≥ (cid:3) Remark 3.5.
There are some remarks on Corollary 3.4.(a) This corollary can be easily extended to the case of a general linear subspaceΛ ≃ P t such that dim X ∩ Λ is zero. Precisely, if Λ does not meet Sing ( X ) and h Λ ∩ X i = Λ, then the t + 1 points of Λ ∩ X are in linearly general position sothat X Λ satisfies N ,p − t − by successive inner projections. To complete thisquestion in Problem (a) for any linear subspace Λ, it remains to consider thathow the projections from a singular center or a linear subvariety containedin X behave (see Question 5.3, 5.5).(b) For a smooth irreducible variety X ⊂ P ( H ( L )) with the condition N p ( p ≥
1) embedded by the complete linear system of a very ample line bundle L on X , Y. Choi, P. Kang and S. Kwak showed that the inner projection X q is smooth and satisfies property N p − for any q ∈ X \ Trisec( X ), i.e.property N p − holds for (Bl q ( X ) , σ ∗ L − E ) by using vector bundle techniquesand Koszul cohomology methods due to Green-Lazarsfeld (see [CKK06]).Our Corollary 3.4 extends this result to the category of reduced projectiveschemes satisfying property N ,p with any smooth point q ∈ X . Note thatthis uniform behavior looks unusual in a sense that linear syzygies of outerprojections heavily depend on moving the center of projection in an ambientspace P N (see [CKP08, KP05, Park08]).In order to understand the Betti table of inner projections, we need to considerdefining equations of inner projections, depth, and the Castelnuovo-Mumford regu-larity. Proposition 3.6. (Quadratic equations of inner projections)
Let X ⊂ P N be a nondegenerate reduced scheme with a defining ideal I and any (possibly singular ) point q ∈ X . For the inner projection X q ⊂ P N − , we have (a) β , ( J ) = β , ( I ) − β , ( K ( I )) , where J is the x -elimination ideal of I asusual. (b) Furthermore, if I is quadratic (so X is quadratic), then we have β , ( J ) = β , ( I ) − N + dim T q X . In particular, in case of I = I X it coincides with h ( P N − , I X q (2)) = h ( P N , I X (2)) − N + dim T q X .
Proof.
As in the proof of Theorem 3.1, there is a long exact sequence such that → Tor S ( I/J, k ) → Tor S ( J, k ) → Tor S ( I, k ) → Tor S ( I/J, k ) → . NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 13
From the reduction of syzygies (2.3) in Remark 2.6, we have ( Tor S ( I/J, k ) ≃ Tor S ( K ( I )( − , k ) = Tor S ( K ( I ) , k ) = 0Tor S ( I/J, k ) ≃ Tor S ( K ( I )( − , k ) = Tor S ( K ( I ) , k ) , which implies the desired formula in (a) directly.If I is a quadratic ideal, then by Proposition 2.5 the K i ( I ) stabilizes at i = 1 and K ( I ) defines the tangent cone T C q X . We may write K ( I ) = ( ℓ , · · · , ℓ t , higher degree polynomials ), where I = ( x ℓ − Q , , · · · , x ℓ t − Q ,t , Q , · · · , Q s ) just as our convention. Amongthe elements of K ( I ) the generators of K ( I ) , { ℓ , · · · , ℓ t } define the tangent space T q X so that we have β , ( K ( I )) = codim( T q X, P N ). (cid:3) Remark 3.7.
In outer projection case, there is a formula h ( I X q (2)) = h ( I X (2)) − ( N − dim Σ q ( X )) if X satisfies property N , (see proposition 4.11 in [AK11], theorem3.3 in [Park08]). This also shows that there is a tendency of having more quadricsfor projected varieties as q is getting closer to X . Note that the negative value of h ( I X q (2)) implies that there is no quadric vanishing on X q . By this fact, we canexpect that the inner projection case has more linear syzygies as Corollary 3.4 shows.Next question is that how many quadrics defining X are required to satisfy prop-erty N ,p and we give the sharp lower bound in the following. Corollary 3.8. (Neccesary lower bound for property N ,p ) Let X be a nonde-generate reduced quadratic scheme in P r + e of codimension e and I be the quadraticideal of X . Suppose that I satisfies property N ,p and β , ( I ) is the number ofgenerators of I . Then β , ( I ) is not less than LB p as follows:LB p = e · p − p ( p − ≤ β , ( I ) ≤ β , ( I X ) (= h ( I X (2)) ) Proof.
Let’s take a smooth point q in X and project X from q . Let X (1) bethe image (the Zariski closure) and I (1) be the elimination ideal of I . Then, fromProposition 3.6 we get β , ( I (1) ) = β , ( I ) − ( r + e ) + r. We also know that I (1) defines X (1) scheme-theoretically and satisfies property N ,p − . Take another smooth point q in X (1) and project it from q . Then, withthe same notation, we have β , ( I (2) ) = β , ( I (1) ) − ( r + e −
1) + r. Taking successive inner projections, we get β , ( I ( p − ) = β , ( I ( p − ) − ( r + e − p + 2) + r. Summing up both sides of above equations, it gives β , ( I ( p − ) = β , ( I ) − ( p − r + 2 e − p + 2)2 + r ( p − · · · ( ∗ ) . And we know that X ( p − is still cut by quadrics (i.e. N , ). So β , ( I ( p − ) is notless than codim X ( p − = ( r + e ) − p + 1 − r = e − p + 1. If we plug-in this inequalityto ( ∗ ), we’ve got the desired bound LB p . (cid:3) Remark 3.9.
This bound is sharp for p = 1 by complete intersections, p = e − del Pezzo varieties (see Theorem 4.3 (b)), and p = e by minimal degree varieties.Note also that the upper bound for β , ( I X ) for a nondegenerate integral subscheme X ⊂ P r + e of codimension e is e ( e +1)2 and this maximum number can be attained ifand only if the variety X is of minimal degree from Corollaries 5 . , . Remark 3.10. (Degree bound by property N ,p ) Recently, A. Alzati and J.C.Sierra get a bound of quadrics for N , as paying attention to the structures of therational map associated to the linear system of quadrics defining X , which coincideswith our bound LB (see [AS10]). They also derive a degree bound in terms ofcodimension e , (cid:0) d (cid:1) ≤ (cid:0) e − e − (cid:1) whose asymptotic behavior is 2 e / √ πe and describe theequality condition: this holds if and only if the equality of LB holds . From thistheorem, in case of p = 2 we also have a rigid condition on degree of the boundary X as if we get some rigidity when p = 1 , e −
1, and e (see Remark 3.9 and Theorem4.3). Using our inner projection method ( e.g. Corollary 3.8), we could improve thisdegree bound a little as follows: (cid:18) d + 2 − p (cid:19) ≤ (cid:18) e + 3 − pe + 1 − p (cid:19) , d ∼ e +2 − p / √ πe (as e getting sufficiently large)under the assumption of property N ,p ( p ≥
2) of X . Example 3.11.
It would be interesting to know that if e ≤ β , ( I X ) < e − X has always at least a syzygy of defining equations which is not linear becauseproperty N , does not hold for X . For example, let C be the general embeddingof degree 19 in P of genus 12. Then C is a smooth arithmetically Cohen-Macaulaycurve which is cut out scheme-theoretically by 9 quadrics, but the homogeneousideal I C is generated by 9 quadrics and 2 cubics (see [Katz93] for details). Thesequadratic generators should have at least a syzygy of higher degree as well as linearsyzygies. Example 3.12. (Veronese embedding v d ( P n ) ) It is shown that v d ( P n ) fails prop-erty N , d − for n ≥ d ≥ v d ( P n ) satisfies property N , d − for n ≥ d ≥ N ,p of Veronese embedding X = v d ( P n ) for some cases by usingthis low bound LB p . For example, when n = 2 , d = 3 , p = 3 d − β , ( I X ) = 27 and LB = 7 · − (cid:0) (cid:1) = 28. Therefore, v ( P ) fails to satisfy N , .Similarly, v ( P ) fails property N , . However, it does not give the reason why v ( P ) does fail to be N , for the case n = d = 3 , p = 3 d − , e = 16, because β , ( I X ) = 126 >
91 = LB . For such p in the middle area of 1 ≤ p ≤ e , LB p seems not to give quite sufficient information for property N ,p , while it may bemore effective to decide N ,p of a given variety for rather large p among 1 ≤ p ≤ e .4. Arithmetic depth and Syzygetic rigidity
Now, we proceed to investigate the depth of inner projections to understandthe shape of the Betti table and Castelnuovo-Mumford regularity. In this sec-tion we always consider the saturated ideal I X among ideals defining X , becausedepth R ( R/I ) = 0 for any defining ideal I which is not saturated. The followingresult looks very surprising when we compare this with the outer projection case(see Remark 4.2). NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 15
Theorem 4.1. (The depth of inner projections)
Let X ⊂ P N be a nonde-generate reduced subscheme and I X be generated by quadrics. Consider the innerprojection π q : X X q ⊂ P N − from a smooth point q ∈ X . Then, (a) the projective dimension of S/I X q , pd S ( S/I X q ) = pd R ( R/I X ) − ; (b) depth R ( X ) = depth S ( X q ) . In particular, X is arithmetically Cohen-Macaulay if and only if so is X q .Proof. (a) We know pd R ( R/I X ) ≥ e = codim X . Let l be pd R ( R/I X ) (so, l ≥ e ),and j =max { j | Tor Rl ( R/I X ) l + j = 0 } . Case 1)
Non-Cohen Macaulay case (i.e. l = e + α, α ≥
1) :First of all, we have the following diagram from the exact sequence (3.1): i of
Tor Si ( S/I X q ) 0 → I X q → I X → I X /I X q → ↑ ↑ ↑ (cid:3) (cid:3) S ( − e ⊕ S ( − e ⊕ · · · ↑ ↑ ↑ e (cid:3) (cid:3) S ( − e − ⊕ S ( − e − ⊕ · · ·↑ ↑ ↑ e + 1 (cid:3) ∼ = (cid:3) l = e + α (cid:4) ∼ = (cid:4) (cid:4) : vanished From this diagram, we get Tor Sl ( R/I X ) ∼ = Tor Sl ( S/I X q ) as finite k -vector spaces.Since Tor Rl +1 ( R/I X ) l +1+ j = 0 for all j ( ∵ pd R ( R/I X ) = l ) and Tor Rl ( R/I X ) l + j = 0for j > j , we can observe using the mapping cone sequence (2.1) that · · · × x ֒ → Tor Sl ( R/I X ) l + j × x ֒ → · · · × x ֒ → Tor Sl ( R/I X ) l + j × x ∼ → · · · So we have Tor Sl ( R/I X ) = 0, because it is finite dimensional. This meansTor Sl ( S/I X q ) ∼ = Tor Sl ( R/I X ) = 0.Next, we claim that Tor Sl − ( S/I X q ) = 0, which implies pd S ( S/I X q ) = l −
1. If α ≥
2, then we have Tor Sl − ( S/I X q ) ≃ Tor Sl − ( R/I X ). Since Tor Sl ( R/I X ) = 0, wehave a nontrivial kernel of the × x map in I X from the mapping cone sequence (2.1)0 → Tor Rl ( R/I X ) l + j ֒ → Tor Sl − ( R/I X ) l − j × x → Tor Sl − ( R/I X ) l − j +1 · · · ( ∗ ) ∦ k ≀ k ≀ Sl − ( I X ) l − j Tor Sl − ( I X ) l − j +1 This implies Tor Sl − ( R/I X ) ≃ Tor Sl − ( S/I X q ) = 0 as wished. So, let us focus onthe case α = 1 and so, l = e + 1. Consider the following sequence and commutativediagram:Tor Sl − ( I X /I X q ) = 0 → Tor Sl − ( I X q ) → Tor Sl − ( I X ) → Tor Sl − ( I X /I X q ) → · · · Tor Sl − ( I X ) e + j ≃ S ( − e − j ) (cid:3) ⊗ k f e + j → S ( − e − j ) ⊗ k ≃ Tor Sl − ( I X /I X q ) e + j h y not injective g y ≀ S ( − c ) ⊗ k ∼ → S ( − c ) ⊗ k where h, g are induced by the multiplication of x n and g is an isomorphism. Tocheck Tor Sl − ( S/I X q ) ∼ = Tor Sl − ( I X q ) = 0, it is enough to show that f : Tor Sl − ( I X ) → Tor Sl − ( I X /I X q ) is not injective because Tor Sl − ( I X /I X q ) = 0. Now let me ex-plain why f be not injective. We get the below isomorphism map for c ≫ Si ( I X q ) are finite-dimensional graded vector spaces and also h is notinjective by ( ∗ ). From Remark (2.6), this diagram commutes and f e + j has anontrivial kernel. Hence f : Tor Sl − ( I X ) → Tor Sl − ( I X /I X q ) is not injective andTor Sl − ( S/I X q ) e + j = Tor Sl − ( I X q ) e + j = 0. Case 2)
Cohen-Macaulay case (i.e. l = e, α = 0) : In this case, we have the longexact sequence on Tor as follows: S ( − e − ⊕ Se ( I X /I X q ) → Tor Se − ( I X q ) → Tor Se − ( I X ) f −→ S ( − e − ⊕· · · Since pd R ( R/I X ) = e , Tor Re +1 ( R/I X ) = 0 and we have an injectionTor Se − ( I X ) e + j = Tor Se ( R/I X ) e + j × x n −→ Tor Se ( R/I X ) e + j + n = Tor Se − ( I X ) e + j + n for any j, n from the mapping cone sequence (2.1). By almost same argument usingthe commuting diagram as Case 1) , α = 1, we can conclude that f : Tor Se − ( I X ) → Tor Se − ( I X /I X q ) is injective and Tor Se ( S/I X q ) = 0. So, this means pd S ( S/I X q ) ≤ e −
1. But we know pd S ( S/I X q ) ≥ codim( X q ) = e −
1, therefore pd S ( S/I X q ) = e −
1. Onthe other hand, (a) implies that depth R ( X ) = depth S ( X q ) by Auslander-Buchsbaumformula. (cid:3) Remark 4.2.
Let X ⊂ P n be a reduced scheme satisfying property N ,p ( p ≥ q ( X ) = { x ∈ X | π q − ( π q ( x )) has length ≥ } is the secant locus of the outerprojection. We would like to remark that depth( X q ) = min { depth( X ) , dim Σ q ( X ) +2 } for a smooth scheme X (see [AK11, Park08]). On the other hand, it would beinteresting to ask the following question: Is there an example such that depth( X q ) =depth( X ) for inner projections?As an interesting application of above results, we can also prove the followingrigidity theorem for the extremal ( i.e. p = e ) and next to extremal ( i.e. p = e − N ,p of the varieties by using Corollary 3.4, Proposition 3.6 andTheorem 4.1. Theorem 4.3. (Syzygetic rigidity on property N ,p ) Let X be a nondegenerate r -dimensional variety (i.e. irreducible, reduced) in P r + e , e = codim( X, P r + e ) . (a) (extremal case) X satisfies property N ,e if and only if X is a minimal degreevariety, i.e. a whole linear space P r + e , a quadric hypersurface, a cone of theVeronese surface in P , or rational normal scrolls; NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 17 (b) (next to extremal case) X fails property N ,e but satisfies N ,e − if and onlyif X is a del Pezzo variety, i.e. X is arithmetically Cohen-Macaulay (abbr.ACM) and is of next to minimal degree.Proof. Let δ ( X ) := deg( X ) − codim( X ) for any subvariety X ⊂ P r + e . Note that δ ( X q ) = δ ( X ) under an inner projection from a smooth point q ∈ X . Take suc-cessive inner projections from smooth points. Call the images (Zariski closure) X = X (0) , X (1) , . . . , X ( e − and I ( i ) for the elimination ideal of I ( i − cutting out X ( i ) scheme-theoretically. By Corollary 3.4 we know that this X ( e − has codim 2and have property N , for (a) (and N , for (b), respectively). Because X ( e − is ahypersurface, by Proposition 3.6 the possible β , ( I ( e − ) = 2 or 3. For the case (a),take an inner projection once more and then X ( e − still satisfies property N , , i.ean (irreducible) quadric hypersurface. So, β , ( I ( e − ) = 1 , β , ( I ( e − ) = 1 + 2 = 3 and δ ( X ) = δ ( X ( e − ) = 1 . That is, X is of minimal degree. In the case of (b), X ( e − is a complete intersectionof two quadrics in P r +2 and X ( e − should be a cubic hypersurface.In particular, the projective dimension of X ( e − is equal to 2 = pd R ( R/I X ) − ( e −
2) by Theorem 4.1. Therefore, β , ( I ( e − ) = 0 , β , ( I ( e − ) = 2 , δ ( X ) = δ ( X ( e − ) = 2 and pd R ( R/I X ) = e, which means X is arithmetically Cohen-Macaulay and of next to minimal degreewith H ( I X (2)) = ( e +2)( e − . By the well-known classification of varieties of next tominimal degree, X is a del Pezzo variety. On the other hand, the curve section C ofa del Pezzo variety X is either an elliptic normal curve or a projection of a rationalnormal curve from a point in Sec( C ) \ C . Since X and C have the same Betti table, X satisfies property N ,e − but fail property N ,e . (cid:3) Remark 4.4.
There are some remarks on Theorem 4.3.(a) For the smooth projectively normal variety X , M. Green’s K p, -theorem in[Gre84] gives a necessary condition on Theorem 4.3 (b) (i.e. X is either avariety of next to minimal degree or a divisor on a minimal degree). Usingour Corollary 3.4 and Depth theorem 4.1, we could obtain the results onboth deg( X ) = codim X + 2 and ACM property and show the rigidity onnext to extremal case for any (not necessarily projectively normal) variety X .(b) Classically, normal del Pezzo varieties were classified by T. Fujita in [Fuj90].And every non-normal del Pezzo variety X (see [BS07, Fuj90]) comes fromouter projection of a minimal degree variety e X from a point q in Sec( e X ) \ e X satisfying dim Σ q ( e X ) = dim e X − q ( e X ) varies as q moves in Sec( e X ) \ e X . Thus one cantry to classify the non-normal del Pezzo varieties by the type of the secantloci Σ q ( e X ). This is recently classified in [BP10] such that there are only 8types of non-normal del Pezzo varieties which are not cones. For example,we find projections of a smooth cubic surface scroll S (1 ,
2) in P from any q ∈ P \ S (1 ,
2) or projections of a smooth 3-fold scroll S (1 , , c ) in P c +4 with c > q ∈ h S (1 , i \ S (1 , , c ), etc. Furthermore, let’s consider the followig category. (We say that an algebraic set X = ∪ X i is connected in codimesion 1 if all the component X i ’s can be arranged insuch a way that every X i ∩ X i +1 is of codimension 1 in X .) { equidimensional, connected in codimension 1, reduced subschemes in P r + e } · · · ( ∗ )In the category ( ∗ ), we have natural notions of dim X and deg( X ) which is thesum of degrees of X i ’s. And as in the category of varieties , we also have the ‘basic’inequality of degree, i.e. deg( X ) ≥ codim X + 1, so it is worthwhile to think of‘minimal degree’ or ‘next to minimal degree’ in this category.Using same methods, the Theorem 4.3 can be easily extended for this category.We call X (or the sequence) linearly joined whenever all the components can beordered X , X , . . . , X k so that for each i , ( X ∪ · · · ∪ X i ) ∩ X i +1 = span( X ∪ · · · ∪ X i ) ∩ span( X i +1 ). Then, we have a corollary as follows: Corollary 4.5.
Let X be a nondegenerate subscheme in the category ( ∗ ) with e =codim( X, P r + e ) . (a) (extremal case) X satisfies property N ,e if and only if X is 2-regular, i.e.the linearly joined squences of r -dimensional minimal degree varieties; (b) (next to extremal case) If X fails property N ,e but satisfies N ,e − , then X is arithmetically Cohen-Macaulay and is of next to minimal degree.Proof. For (a) we can also project X to a hyperquadric X ( e − similarly (In thiscase X ( e − is reducible, i.e. a union of two r -linear planes and every component of X degenerates into this linear subspaces). So X ( e − is ACM, and from our Depththeorem 4.1 X is also ACM, eventually 2-regular. We also have a similar result forthe case of (b) by same arguments; X becomes ACM and of next to minimal degreesubscheme with h ( I X (2)) = ( e +2)( e − in this category. (cid:3) Remark 4.6.
For the ‘rigidity’ of property N ,p for p = e , D. Eisenbud et al. provedthe same theorem for the category of algebraic set more generally in [EGHP05]. The(geometric) classification of 2-regularity is well-known for varieties, and for generalalgebraic sets it is given in [EGHP06]. We reprove this rigidity (case (a)) usingour inner projection method and for next to extremal case (b) we also get similarclassification for the subschemes in the category ( ∗ ) (see Question 5.6).5. Examples and Open questions
It seems to be quite natural to find out a good inner projection as we move thepoint q ∈ X in many aspects. What happens to inner projections from singularpoints? During the discussions with P. Schenzel, we have the following example; Example 5.1. (Projection from a singular point, discussion with P. Schenzel)Let us consider a singular surface X in P by Segre embedding of quadric in P andsingular quintic rational curve in P (Note that ‘-’ means “zero” in the Betti table); NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 19 · · · i · · · − − − − − · · · − · · · −
70 475 1605 3333 4500 · · · β i, · · · − − −
11 100 405 · · · β i, · · · − − − − − − . . . β i, . . .(5.1) Table 5.1
A singular surface X in P (computed by Singular )We see that X satisfies property N , . Now consider inner projections of X from (a) asmooth point and (b) any singular point of X (we can’t distinguish the singularities).( a ) 0 1 2 3 40 1 − − − − −
58 351 1035 · · · − − · · · − − − − . . . ( b ) 0 1 2 3 40 1 − − − − −
59 362 1089 · · · − − − · · · − − − − . . .(5.2) Table 5.2 (a) from smooth point, (b) from any singular pointWhile X q property N , holds in case (a) as our Corollary 3.4 says, in case (b) westill have property N , for X q ! Example 5.2.
Let X be the Grassmannian G (2 ,
4) in P , 6- dimensional del Pezzovariety of degree 5 whose Betti table is0 1 2 30 1 − − − − − − − − Table 5.3 the Grassmannian G (2 ,
4) in P and property N , is satisfied. Since it is homogeneous and covered by lines, sowe can choose any (smooth) point q in X and a line ℓ through q in X . Then theprojection X q is a complete intersection of two quadrics in P (property N , ) and q ′ = π q ( ℓ ) becomes a singularity of multiplicity 2 in X q . If we project this onemore from q ′ , then the projected image becomes a quadric hypersurface in P stillsatisfying property N , . Question 5.3. (Inner projection from a singular point) Assume that X be a non-degenerate projective scheme with N ,p . If q ∈ X is singular, we could expect thatthe inner projection from q has more complicate aspects, but shows better behaviorstill satisfying N ,p in many experimental data. What can happen to the projectionfrom singular locus in general?Next, we consider Problem (a) of the introduction in general. Let X be a non-degenerate subscheme with property N ,p . If ℓ meets X but is not contained in X , then we can regard the projection π ℓ as the composition of two simple projec-tions from points q , q . Furthermore, if such ℓ meets X at smooth two points, then X ℓ = π ℓ ( X \ ℓ ) satisfies property N ,p − by our main Theorem.But not the case of ℓ ⊂ X we can treat simply, because q = π q ( ℓ ) might be asingular point even if π ℓ = π q ◦ π q . In this case, we give an interesting exampleshowing that the Betti numbers of X ℓ = π ℓ ( X \ ℓ ) are related to the geometry ofthe line in X . Example 5.4. (Projection from a line inside the variety) Consider the Segre em-bedding X = σ ( P × P ), 6-fold of degree 15 in P having property N , whoseBetti table is 0 1 2 3 4 5 6 7 80 1 − − − − − − − − −
30 120 210 168 50 − − − − − − −
50 120 105 40 6 . (5.4) Table 5.4
Segre embedding X = σ ( P × P ), 6-fold of degree 15 in P In X there are two type of contained lines, so called ℓ and ℓ . If we take ℓ as the line σ ( { pt } × ℓ ) in X , then the image X ℓ is the intersection of two cones h σ ( P × P ) , P i and h P , σ ( P × P ) i which is a 6-fold of degree 12 in P satisfying property N , with the Betti table as in Table 5.5 (a).On the other hand, in case of the line ℓ = σ ( ℓ × { pt } ), X ℓ is a 6-dimensionalcone h{ pt } , σ ( P × P ) i of degree 10 in P and has its own Betti table as in Table 5.5(b) with property N , . Note that the dimension of the span h∪ q ∈ ℓ T q X i of tangentspaces along ℓ is 8, but dim h∪ q ∈ ℓ T q X i = 10 (i.e. the tangent spaces change morevariously along ℓ than ℓ ). So, it is expected that ℓ is geometrically less movablethan ℓ inside X and X ℓ has more linear syzygies.( a ) 0 1 2 3 4 5 60 1 - - - - - -1 - 16 40 30 4 - -2 - - - 20 40 24 5 ( b ) 0 1 2 3 4 5 60 1 - - - - - -1 - 18 52 60 24 - -2 - - - - 10 12 3(5.5) Table 5.5 (a) from a line ℓ of type 1, (b) from a line ℓ of type 2 Question 5.5. (Inner projection from a subvariety) Let X be a nondegeneratereduced scheme in P N satisfying property N ,p ( p >
1) which is not necessarily
NALYSIS ON SOME INFINITE MODULES, INNER PROJECTION, AND APPLICATIONS 21 linearly normal. Consider the inner projection from a line ℓ ⊂ X . Is it true that π ℓ ( X \ ℓ ) satisfies at least N ,p − ? How does the infinitesimal geometry of ℓ in X effect to the syzygies of π ℓ ( X \ ℓ )? More generally, how about the projection froma subvariety Y of X ? The projection from Y is defined by the projection fromΛ := h Y i , the linear span of Y (see [BHSS00]). Say dim Λ = t < p . Does X Λ in P N − t − satisfy property N ,p − t − in general as raised in the problem list (a) in theintroduction?For the sake of Question 5.5, we expect to need developing the elimination map-ping cone theorem and partial elimination module theory for multivariate case andcalculating on the syzygies of those partial elimination modules by Gr¨obner basistheory for graded modules. See [HK10] for basic settings and some partial resultsfor bivariate case.Finally, we have the following question as rasied in Remark 4.6. Question 5.6. (Geometric characterization of some 3-regular ACM schemes) Weshowed in Section 4 that if a r -equidimensional, reduced and connected in codi-mension 1 subscheme X in P r + e fails property N ,e but satisfies N ,e − , then it isa ACM, 3-regular scheme of next to minimal degree (i.e. deg( X ) = codim X + 2)with h ( I X (2)) = ( e +2)( e − . Further, a theorem of L.T. Hoa in [Hoa93] gives thecomplete graded Betti numbers of these schemes as follows:0 → R ( − e − → R β e − , ( − e ) → R β e − , ( − e +1) → · · · → R β , ( − → I X → · · · ( ∗∗ )where β i, = ( i + 1) (cid:0) e +1 i +2 (cid:1) − (cid:0) ei (cid:1) for 0 ≤ i ≤ e −
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Algebraic Structure and its Applications Research Center(ASARC), Departmentof Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-Gu, Daejeon, Korea
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