aa r X i v : . [ m a t h . AG ] J u l On the Arithmetical Rank of Certain SegreEmbeddings
Matteo Varbaro
Dipartimento di MatematicaUniversit`a degli Studi di Genova, Italy [email protected]
October 27, 2018
Abstract
We study the number of (set-theoretically) defining equations of Segre products of projec-tive spaces times certain projective hypersurfaces, extending results by Singh and Walther.Meanwhile, we prove some results about the cohomological dimension of certain schemes.In particular, we solve a conjecture of Lyubeznik about an inequality involving the cohomo-logical dimension and the ´etale cohomological dimension of a scheme, in the characteristic-zero-case and under a smoothness assumption. Furthermore, we show that a relationshipbetween depth and cohomological dimension discovered by Peskine and Szpiro in positivecharacteristic holds true also in characteristic-zero up to dimension three.
The beauty of find the number of defining equations of a variety is expressed by Lyubeznik in[Ly2] as follows:
Part of what makes the problem about the number of defining equations so interesting isthat it can be very easily stated, yet a solution, in those rare cases when it is known, usually ishighly non trivial and involves a fascinating interplay of Algebra and Geometry.
In this paper we study the number of defining equations, called arithmetical rank (see Sec-tion 2), of certain Segre products of two projective varieties. Let us list some works that therealready exist in this direction.1. In their paper [BS], Bruns and Schw¨anzl studied the number of defining equations of adeterminantal variety. In particular they proved that the Segre product P n × P m ⊆ P N ,where N = nm + n + m , can be defined set-theoretically by N − n = m = E × P ⊆ P where E is a smooth elliptic plane curve: the authors proved that the arithmetical rank of thisSegre product is 4. Later, in [So], Song proved that the arithmetical rank of C × P , where C ⊆ P is a Fermat curve (i.e. a curve defined by the equation x d + x d + x d ), is 4. Inparticular both E × P and C × P are not set-theoretic complete intersections.1n light of these results it is natural to study the following problem. Let n , m , d be natural numbers such that n ≥ and m , d ≥ , and let X ⊆ P n be a smoothhypersurface of degree d. Consider the Segre product Z = X × P m ⊆ P N , where N = nm + n + m.What can we say about the number of defining equations of Z? Notice that the arithmetical rank of Z can depend, at least a priori, by invariant different from n , m , d : in fact we will need other conditions on X . However for certain n , m , d we can providesome answers to this question. To this aim we will use various arguments: from complexanalysis to the theory of algebras with straightening law, passing through local cohomology,´etale cohomology and much commutative algebra.In the case n = m =
1, we introduce, for every d , a locus of special smooth projec-tive plane curves of degree d , that we will denote by V d : this locus consists in those smoothprojective curves C of degree d which have a d -flex, i.e. a point P at which the intersectionmultiplicity of C and the tangent line in P is equal to d . Using methods coming from “algebraswith straightening law’s theory” we prove that for such a curve C the arithmetical rank of theSegre product C × P ⊆ P is 4, provided that d ≥ V and that every Fermat’s curve of degree d belongs to V d , so we recover the results obtained in [SW] and in [So]. However the equations that we willfind are different from those found in these papers, and our result is characteristic free. Notethat a result of Casnati and Del Centina [CD] shows that the codimension of V d in the locus ofall the smooth projective plane curves of degree d is d −
3, provided that d ≥ n , we can prove that if X ⊆ P n is a general smooth hypersurface of degree notbigger than 2 n −
1, then the arithmetical rank of X × P ⊆ P n + is at most 2 n (Corollary 3.8).To establish this we need a higher-dimensional version of V d . This result is somehow in thedirection of the open question whether any connected projective scheme of positive dimensionin P N can be defined set-theoretically by N − F = x dn + (cid:229) n − i = x i G i ( x , . . . , x n ) and X = V + ( F ) is smooth, then the arithmetical rank of X × P ⊆ P n + is 2 n − C × P m ⊆ P m + , where C is a smooth conic of P , is equal to3 m , provided that char ( k ) = C × P m is a set-theoretic completeintersection if and only if m = k = C and using singular, local and sheaves cohomology.The results of Section 2 yield some nice consequences, independently from Section 3:1. For any n , m and d , if X is smooth, the arithmetical rank of X × P m ⊆ P N can vary justamong N − N − N .2. A conjecture of Lyubeznik in [Ly4] (see Conjecture 2.20) states, roughly speaking, that“the `etale cohomology provides a better lower bound for the arithmetical rank than the lo-cal cohomology”. We prove the conjecture in the characteristic 0 case under a smoothness2ssumption, see Theorem 2.21.3. We extend a result by Speiser obtained in characteristic 0 in [Sp], regarding the arithmeti-cal rank of the diagonal in P n × P n , to any characteristic, see Corollary 2.14.4. As a consequence of Theorem 2.18, we get that if a smooth projective surface X has aCohen-Macaulay homogeneous coordinate ring, then the cohomological dimension of itscomplement in any P n is the least possible (codim P n X − As already said in the introduction, in this section we will get the necessary lower boundswe need using results about the cohomological dimension of open subschemes of projectiveschemes.First we describe in a precise way the setting in which we will work: for a noetherian ring R and an ideal I ⊆ R we define the arithmetical rank of I with respect to R as the integerara ( I ) = min { k : ∃ f , . . . , f k ∈ R such that √ I = p ( f , . . . , f k ) } . Notice that to be more precise we should write ara R ( I ) , however it will be always clear from thecontext who is R . A lower bound for the arithmetical rank is given by Krull’s Hauptidealsatz:ara ( I ) ≥ ht ( I ) . If R is graded and I homogeneous we can also define the homogeneous arithmetical rank, thatis the integerara h ( I ) = min { k : ∃ f , . . . , f k ∈ R homogeneous such that √ I = p ( f , . . . , f k ) } . Obviously we have ara ( I ) ≤ ara h ( I ) . Assume that R is a polynomial ring of N + k , and that I is a homogeneousideal of R . Then ara ( I ) gives the least number of hypersurfaces of the affine space A N + todefine set-theoretically V ( I ) = { ˆ ∈ Spec ( R ) : ˆ ⊇ I } ⊆ A N + = Spec ( R ) ; similarly ara h ( I ) gives the least number of hypersurfaces of P N to intersect set-theoretically to obtain V + ( I ) = { ˆ ∈ Proj ( R ) : ˆ ⊇ I } ⊆ P N = Proj ( R ) . It is an open problem whether these two numbers arealways equal (see the survey article of Lyubeznik [Ly1]).3 emark 2.1. The reader should be careful to the following: the number ara ( I ) , where I isan ideal of a polynomial ring, in general, does not give the minimal number of polynomialswhose zero-locus is the same zero-locus of I , namely Z ( I ) . For instance, if I = ( f , . . . , f m ) ⊆ R [ x , . . . , x N ] , clearly Z ( I ) = Z ( f + . . . + f m ) . However ara ( I ) can be bigger than 1. The reader should keep in mind that, unless the base fieldis algebraically closed, there is no relations between V ( I ) and Z ( I ) .We will say that I (or X = V + ( I ) ) is a set-theoretic complete intersection if ara h ( I ) = ht ( I ) = codim P N X .For a Noetherian ring R and an ideal I ⊆ R the cohomological dimension cd ( R , I ) of I (withrespect to R ) is the supremum of the integers i such that there exists an R -module M for which H iI ( M ) =
0. It is well known that ara ( I ) ≥ cd ( R , I ) ≥ ht ( I ) . In the same way, the cohomological dimension cd ( X ) of a scheme X is the supremum integer i such that there exists a quasi coherent sheaf F such that H i ( X , F ) = R is a finitely generated positively graded k -algebra and I ⊆ R is a homogeneous ideal non-nilpotent, then cd ( R , I ) − = cd ( Spec ( R ) \ V ( I )) = cd ( Proj ( R ) \ V + ( I )) (1)(see Hartshorne [Har1]); so to bound the arithmetical rank of I , and hence the homogeneousarithmetical rank, we will give bounds on cd ( Proj ( R ) \ V + ( I )) . Throughout this subsection k (or K ) will denote a field of characteristic 0. The following remarkallows us to can assume, in many cases, that the base field is C . Remark 2.2.
Let R be an A -algebra, a ⊆ R an ideal, B a flat A -algebra, R B = R ⊗ A B , M an R -module and M B = M ⊗ A B . Using the ˘Cech complex it is not difficult to prove that for every j ∈ N : H j a ( M ) ⊗ A B ∼ = H j a R B ( M B ) (2)Now let S = K [ x , . . . , x n ] and I ⊆ S an ideal. Since I is finitely generated we can find a field k such that, setting S k = k [ x , . . . , x n ] , the following properties hold: k ⊆ K , Q ⊆ k ⊆ C , ( I ∩ S k ) S = I (to this aim we only have to add to Q the coefficients of a set of generators of I ). Since K and C are k -algebras faithfully flat equation (2) implies thatcd ( S , I ) = cd ( S k , I ∩ S k ) = cd ( S C , ( I ∩ S k ) S C ) , (3)where S C = C [ x , . . . , x n ] .In the above situation assume that I is homogeneous and that X = Proj S / I is smooth over K .Then set X k = Proj ( S k / ( I ∩ S k )) and X C = Proj ( S C / (( I ∩ S k ) C )) . Notice that X ∼ = X k × k Spec K ,4 C ∼ = X k × k Spec C , and that X k (respective X C ) is smooth over k (respective over C ). By basechange (see Liu [Li, Chapter 6, Proposition 1.24 (a)]) and by the fact that K and C are both flat k -algebras, we get, for all natural numbers i , j , H i ( X , W jX / K ) ∼ = H i ( X k , W jX k / k ) ⊗ k K and H i ( X C , W jX C / C ) ∼ = H i ( X k , W jX k / k ) ⊗ k C (see [Li, Chapter 5, Proposition 2.27]). Particularly we havedim K ( H i ( X , W jX / K )) = dim C ( H i ( X C , W jX C / C )) (4)In the rest of this subsection k will denote a field of characteristic 0. Moreover, if X is aprojective variety smooth over k , we will write h i j ( X ) for dim k ( H i ( X , W jX / k )) .In the next remark, for the convenience of the reader, we collect some well known factswhich we will use throughout the paper. Remark 2.3.
Let X be a projective scheme over C : we will denote b i ( X ) the topological Bettinumber b i ( X ) = rank Z ( H Singi ( X an , Z )) = rank Z ( H iSing ( X an , Z )) == dim C ( H iSing ( X an , C )) = dim C ( H i ( X an , C )) ( X an means X regarded as a complex manifold, in the sense of Serre [Se], and C denotes thelocally constant sheaf associated to C ). Pick another projective scheme over C , say Y , anddenote by Z the Segre product X × Y . The K¨unneth formula for singular cohomology (forinstance see Hatcher [Hat, Theorem 3.16]) yields H iSing ( Z an , C ) ∼ = M p + q = i H pSing ( X an , C ) ⊗ C H qSing ( Y an , C ) , thus b i ( Z ) = (cid:229) p + q = i b p ( X ) b q ( Y ) . (5)Now assume that X is a projective variety smooth over C . It is well known that X an is aK¨ahler manifold, so the Hodge decomposition (see the notes of Arapura [Ar, Theorem 10.2.4])is available. Therefore together with a theorem of Serre (see [Se, Theoreme 1, pag. 19]) wehave H iSing ( X an , C ) ∼ = M p + q = i H p ( X an , ( W X / C ) qan ) ∼ = M p + q = i H p ( X , W qX / C ) , where F an is the analyticization of a sheaf F (see [Se]). Thus b i ( X ) = (cid:229) p + q = i h pq ( X ) (6)Finally note that the restriction map on singular cohomology H iSing ( P nan , C ) −→ H iSing ( X an , C ) (7)is injective provided that i = , . . . , X (see Shafarevich [Sh, pp. 121-122]). In particular,since b i ( P n ) = i ≤ n , it follows that b i ( X ) ≥ i ≤ dim X (8)5he following theorem is a quite simple consequence of the results of [Og]. It providessome necessary and sufficient conditions for the cohomological dimension of the complementof a smooth variety in a projective space to be smaller than a given integer. Theorem 2.4.
Let X ⊆ P n be a projective variety smooth over k, r an integer greater than orequal to codim P n X and U = P n \ X . Then cd ( U ) < r if and only ifh pq ( X ) = (cid:26) if p = q , p + q < n − r if p = q , p + q < n − rMoreover, if k = C , the above conditions are equivalent to: b i ( X ) = (cid:26) if i < n − r and i is even if i < n − r and i is oddProof. By equations (3) and (4) of Remark 2.2 we can reduce the problem in the case in which k = C . So the “only if”-part follows by a result of Hartshorne [Har2, Corollary 7.5, p. 148].So it remains to prove the “if”-part. By a theorem of Grothendieck in [Gr1] algebraic DeRham cohomology agrees with singular cohomology. Therefore by the last part of Remark 2.3the restriction maps H iDR ( P n ) −→ H iDR ( X ) (9)(where H DR denotes the algebraic De Rham cohomology) are injective for all i ≤ X . Bythe assumptions, equation (6) yields b i ( X ) = i is even and i < n − r , 0 otherwise. Moreover b i ( P n ) = i is even and i ≤ n , 0 otherwise. So using again the result of Grothendieck themaps in (9) are isomorphisms for all i < n − r .Now we would use a result of Ogus ([Og, Theorem 4.4]), and to this aim we will show thatthe De Rham-depth of X is greater than or equal to n − r . By the proof of [Og, Theorem 4.1]this is equivalent to the fact that Supp ( H i a ( S )) ⊆ m for all i > r , where S = C [ x , . . . , x n ] , a ⊆ S isthe ideal defining X and m is the maximal irrelevant ideal of S . But this is easy to see, becauseif ˆ is a graded prime ideal containing a and different from m , being X non singular, a S ˆ isa complete intersection in S ˆ : so ( H i a ( S )) ˆ ∼ = H i a S ˆ ( S ˆ ) = i > r ( ≥ ht ( a S ˆ )) . Hence[Og, Theorem 4.4] yields the conclusion.Finally, if k = C , the last condition is a consequence of the first one by equation (6). More-over, it implies the first one because the restriction maps of singular cohomology H iSing ( P nan , C ) −→ H iSing ( X an , C ) (that are injective if i < n − r by the last part of Remark 2.3) are compatible with the Hodgedecomposition (see [Ar, Corollary 11.2.5]). Remark 2.5.
Theorem 2.4 does not hold in positive characteristic: for instance pick an ellipticcurve E over a field of positive characteristic whose Hasse invariant is 0. Then set X = E × P ⊆ P and U = P \ X . The Frobenius acts as 0 on H ( X , O X ) , so cd ( U ) = H ( X , O X ) = roposition 2.6. Let X and Y be two positive dimensional projective schemes smooth over k,and set Z = X × Y ⊆ P N (any embedding) and U = P N \ Z. Then cd ( U ) ≥ N − . In particularif dim Z ≥ , Z is not a set-theoretic complete intersection.Proof. By equation (3) we can assume k = C . Using equation (8) we have b ( X ) ≥ b ( X ) ≥ b ( Y ) ≥ b ( Y ) ≥
1, so equation (5) implies b ( Z ) ≥
2. Now equation (6) and Theorem2.4 yield the conclusion.
Remark 2.7.
The proof of Proposition 2.6 yields the following nice fact:Let X and Y be two positive dimensional projective varieties smooth over C and Z = X × Y ⊆ P N . Then the dimension of the secant variety of Z in P N is at least 2 dim Z − b ( Z ) ≥
2. By a theorem of Barth(see Lazarsfeld [La, Theorem 3.2.1]), it follows that Z cannot be embedded in any P M with M < X −
1. If the dimension of the secant variety were less than 2 dim X −
1, it wouldbe possible to project down in a biregular way X from P N in P X − , and this would be acontradiction.Note that the above lower bound is the best possible, in fact P r × P s can be embedded in P ( r + s ) − (see Hartshorne [Har3, p. 1026]). Remark 2.8.
The statement of Proposition 2.6 is false in positive characteristic. To see this,consider two Cohen-Macaulay graded k -algebras A and B of negative a -invariant. Set R = A B their Segre product (with the notation of the paper of Goto and Watanabe [GW]). By [GW,Theorem 4.2.3] R is Cohen-Macaulay as well. So, presenting R as a quotient of a polynomialring of N + R ∼ = P / I , a theorem of Peskine and Szpiro in [PS] implies thatcd ( P , I ) = N + − dim R (because char ( k ) > X = Proj ( A ) , Y = Proj ( B ) , Z = Proj ( R ) ⊆ P N = Proj ( P ) and cd ( P N \ Z ) = cd ( P , I ) − = N − dim Z − Proposition 2.9.
Assume that X is a projective variety smooth over k such that H ( X , O X ) = and let Y be any projective scheme over k. As above set Z = X × Y ⊆ P N (any embedding) andU = P N \ Z. Then cd ( U ) ≥ N − .Proof. By virtue of Remark 2.2 we may assume k = C . The assumptions imply that b ( X ) = b ( Z ) = U is smooth, so [Har2, Theorem 7.4,p. 148] implies the conclusion. Remark 2.10.
If in the situation of proposition 2.9 dim Z ≥
2, then it follows that Z cannotbe a set–theoretical complete intersection. This is a consequence of a more general result ofHartshorne obtained in [Har1], that states that an irregular projective variety X over a fieldof characteristic 0 (i.e. q ( X ) = h ( X ) = P n . If the characteristic of the base field is 0 we have seen in the previous subsection that we can,usually, reduce the problem to k = C ; in this context is available the complex topology, so wecan use methods from algebraic topology and from complex analysis.7nfortunately when the characteristic of k is positive, the above techniques are not available.Moreover some of the results obtained in Subsection 2.1 are not true in positive characteristic,as we have shown in Remarks 2.5 and 2.8. To avoid these difficulties we have to use ´etale coho-mology, that gives a lower bound for the number of equations defining a variety as well as localcohomology (see equation (10) of Remark 2.11). This subject was introduced by Grothendieckin [Gr3]. Other references are the book [Mi1] and the lectures [Mi2] of Milne.For a scheme X we denote by X ´ et the ´etale site of X and, with a slight abuse of notation, by Z / l Z the constant sheaf associated to Z / l Z (for any l ∈ Z ). Moreover, we denote by ´ecd ( X ) the´etale cohomological dimension of X , that is the largest integer i such that there exists a torsionsheaf F on X ´ et with ´etale cohomology group H i ( X ´ et , F ) = H i denotes the usual cohomologyof sheaves). Below we collect some basic results about the ´etale cohomological dimension. Remark 2.11. If X is a n -dimensional scheme of finite type over a separably closed field, then´ecd ( X ) ≤ n ([Mi1, Chapter VI, Theorem 1.1]). If moreover X is affine, then ´ecd ( X ) ≤ n ([Mi1,Chapter VI, Theorem 7.2]).Assume that X = Proj ( R ) is projective and pick a closed subscheme Y = V + ( I ) ⊆ X . Then U = X \ Y can be cover by ara h ( I ) affine subsets of X . Moreover the ´etale cohomologicaldimension of these affine subsets is less than or equal to n for what said above. So, usingrepetitively the Mayer-Vietoris sequence ([Mi1, Chapter III, Exercise 2.24]), it is easy to provethat ´ecd ( U ) ≤ n + ara h ( I ) − Theorem 2.12. (Lyubeznik) Let k be a separably closed field of arbitrary characteristic, Y ⊆ Xtwo projective varieties such that U = X \ Y is non-singular. Set N = dim X , and l ∈ Z coprimewith the characteristic of k. If ´ecd ( U ) < N − r, then the restriction mapsH i ( X ´ et , Z / l Z ) −→ H i ( Y ´ et , Z / l Z ) are isomorphism for i < r and injective for i = r. Remark 2.13.
The ´etale version of Theorem 2.4 does not hold. In fact, the integer ´ecd ( P N \ Y ) is not an invariant of only Y and N (as instead is for the integer cd ( P N \ Y ) ). For instance we canconsider P ⊆ P (embedded as a linear subspace) and v ( P ) ⊆ P (where v ( P ) is the 2ndVeronese embedding): the first one is defined (also scheme-theoretically) by 3 linear equations,so ´ecd ( P \ P ) ≤ ( P \ v ( P )) = X ⊆ P n depends on the embedding, and not only on X and on n . This suggests thelimits of the use of local cohomology on certain problems regarding the arithmetical rank.In [Sp] Speiser, among other things, computed the arithmetical rank of the diagonal D = D ( P n ) ⊆ P n × P n , provided that the characteristic of the base field is 0. In characteristic p > P n × P n \ D is the least possible, i.e. n − D . Actually Theorem8.12 easily implies that the result of Speiser holds in arbitrary characteristic, since the upperbound found in [Sp] is valid in arbitrary characteristic. However, since in that paper the authordid not describe the equations needed to define set-theoretically D , we provide the upper boundwith a different method, that yields an explicit set of equations for D .To this aim, we recall that the coordinate ring of P n × P n is A = k [ x i y j : i , j = , . . . , n ] andthe ideal I ⊆ A defining D is I = ( x i y j − x j y i : 0 ≤ i < j ≤ n ) . Corollary 2.14.
In the situation described above (with k a separably closed field of arbitrarycharacteristic) ara h ( I ) = n − .Proof. As already said, by [Sp, Proposition 2.1.1] we already know that ara h ( I ) ≤ n −
1. How-ever we can observe that, if we consider IR ⊆ R = k [ x i , y j : i , j = , . . . , n ] , then IR is the idealgenerated by the 2-minors of the 2 × ( n + ) matrix of indeterminates whose rows are, respec-tively, x , . . . , x n and y , . . . , y n . So, by [BrVe, (5.9) Lemma], a set of generators of IR ⊆ R up toradical is g k = (cid:229) ≤ i < j ≤ ni + j = k ( x i y j − x j y i ) , k = , . . . , n − . Since these polynomials belong to A and since A is a direct summand of R , we get p ( g , . . . , g n − ) A = I , therefore ara h ( I ) ≤ n − l coprime with char ( k ) . K¨unneth formula for ´etale cohomology([Mi1, Chapter VI, Corollary 8.13]) implies that H ( P n ´ et × P n ´ et , Z / l Z ) ∼ = ( Z / l Z ) , while H ( D ´ et , Z / l Z ) ∼ = H ( P n ´ et , Z / l Z ) ∼ = Z / l Z . So Theorem 2.12 yields ´ecd ( U ) ≥ n −
2, where U = P n × P n \ D . Therefore equation (10) yields the conclusion.The next two propositions are the analogue of Propositions 2.6 and 2.9. We need them tocompute the homogeneous arithmetical rank of certain Segre products in arbitrary characteristic.First we need a remark: Remark 2.15.
Let X be a projective variety smooth over a field k and l an integer coprimeto char ( k ) . The kernel of the cycle map is contained in the kernel of the projection from theChow ring to itself modulo numerical equivalence. But this last group is non-zero because X isprojective, so we have H i ( X ´ et , Z l ) = ∀ i = , . . . , dim X . (11)Therefore there exists an integer n such that H i ( X ´ et , Z / l n Z ) is non-zero for any i = , . . . , dim X .See the proof of [Mi1, Chapter VI, Theorem 11.7]. Proposition 2.16.
Let k an algebraic closed field of arbitrary characteristic. Let X and Ybe two projective varieties smooth over k of dimension at least 1. Set Z = X × Y ⊆ P N (anyembedding) and U = P N \ Z. Then ´ecd ( U ) ≥ N − . In particular if dim Z ≥ , Z is not aset-theoretic complete intersection. roof. By the above remark there is an integer l coprime with char ( k ) such that the mod-ules H i ( X ´ et , Z / l Z ) and H i ( Y ´ et , Z / l Z ) are non-zero Z / l Z -modules. But H ( P N ´ et , Z / l Z ) ∼ = Z / l Z ,therefore by K¨unneth formula for ´etale cohomology ([Mi1, Chapter VI, Corollary 8.13]) it fol-lows that H ( Z ´ et , Z / l Z ) cannot be isomorphic to H ( P N ´ et , Z / l Z ) . Now Theorem 2.12 impliesthe conclusion. Proposition 2.17.
Let k an algebraically closed field, C a smooth projective curve of positivegenus, X a projective scheme and Y = C × X ⊆ P N (any embedding). Then ´ecd ( P N \ Y ) ≥ N − . In particular, if dim X ≥ , then Y is not a set-theoretic complete intersection.Proof. Set g the genus of C . By [Mi2, Proposition 14.2 and Remark 14.4] it follows that H ( C ´ et , Z / l Z ) ∼ = ( Z / l Z ) g . Moreover H ( X ´ et , Z / l Z ) = H ( P N ´ et , Z / l Z ) =
0. But byK¨unneth formula for ´etale cohomology H ( Y ´ et , Z / l Z ) =
0, therefore Theorem 2.12 let us con-clude.
In this subsection we draw two nice consequences of the investigations we made in the first partof the work. They are in the direction of a problem stated by Grothendieck, who asked in [Gr2,p. 79] to find conditions, fixed a positive integer c , under which cd ( R , I ) ≤ c , where I is an idealin a ring R .The first fact we want to present is a consequence of Theorem 2.4, and regards a relation-ship between cohomological dimension of an ideal in a polynomial ring and the depth of therelative quotient ring. It was proved by Peskine and Szpiro in [PS] that if I ⊆ S = k [ x , . . . , x n ] is a homogeneous ideal of a polynomial ring over a field of positive characteristic such thatdepth ( S / I ) ≥ t , then cd ( S , I ) ≤ n − t . The same assertion does not hold in characteristic 0,in fact examples are known already for t = I defines the Segre product oftwo projective spaces). When t = t = Theorem 2.18.
Let S = k [ x , . . . , x n ] be a polynomial ring over a field of characteristic . If I ⊆ Sis a homogeneous prime ideal such that ( S / I ) ˆ is a regular local ring for any homogeneousprime ideal ˆ = m = ( x , . . . , x n ) and such that depth ( S / I ) ≥ , then cd ( S , I ) ≤ n − .Proof. Suppose by contradiction that cd ( S , I ) ≥ n −
2. Set X = Proj ( S / I ) ⊆ P n − = Proj ( S ) . Sowe are supposing that cd ( P n − \ X ) ≥ n − X is a projectivevariety smooth over k , therefore Theorem 2.4 implies that h ( X ) = h ( X ) =
0. Butwith the notation of Remark 2.2, h ( X ) = h ( X C ) and h ( X ) = h ( X C ) . So, since h ( X C ) = h ( X C ) (using [Ar, Theorem 10.2.4] and [Se] together), we have h ( X ) =
0. But H ( X , O X ) =[ H m ( S / I )] ⊆ H m ( S / I ) ( [ ] denotes the 0-degree part), so depth ( S / I ) ≤
2, that is a contradiction.Actually the cited result of Peskine and Szpiro holds true whenever the ambient is a regularlocal ring of positive characteristic. Moreover, one can easily deduce by the result of Hunekeand Lyubeznik [HL, Theorem 2.9] the following: If R is an n -dimensional regular local ringcontaining its residue field and a ⊆ R is an ideal such that depth ( R / a ) ≥
2, then cd ( R , a ) ≤ n − uestion 2.19. Suppose that R is a regular local ring, and that I ⊆ R is an ideal such that depth ( R / I ) ≥ . Is it true that cd ( R , I ) ≤ dim R − ? The second fact we want to show is a consequence of Theorem 2.4 and Theorem 2.12. Itprovides a solution of a special case of a conjecture stated by Lyubeznik in [Ly4, Conjecture, p.147]:
Conjecture 2.20. (Lyubeznik) If U is a n-dimensional scheme of finite type over a separablyclosed field, then ´ecd ( U ) ≥ n + cd ( U ) . Theorem 2.21.
Let X ⊆ P n be a projective variety smooth over C , and U = P n \ X . Then ´ecd ( U ) ≥ n + cd ( U ) Proof.
Set cd ( U ) = s , and define an integer r s to be 0 (resp. 1) if n − s − n − s − b n − s − ( X ) > r s .Consider, for a prime number p , the Z / p Z -vector space Hom Z ( H Singi ( X an , Z ) , Z / p Z ) . Since H Singi ( X an , Z ) is of rank bigger than r s , the above Z / p Z -vector space has dimension greaterthan r s . Therefore by the surjection given by the universal coefficient theorem H n − s − Sing ( X an , Z / p Z ) −→ Hom Z ( H Singn − s − ( X an , Z ) , Z / p Z ) (see [Hat, Theorem 3.2, p. 195]), we infer that dim Z / p Z H n − s − Sing ( X an , Z / p Z ) > r s . Now acomparison theorem due to Grothendieck (see [Mi1, Chapter III, Theorem 3.12]) yieldsdim Z / p Z H n − s − ( X ´ et , Z / p Z ) > r s . Since dim Z / p Z ( H n − s − ( P n ´ et , Z / p Z )) = r s , Theorem 2.12 implies that ´ecd ( U ) ≥ n − ( n − s ) = n + s .Theorem 2.21 might look like a very special case of Conjecture 2.20. However the casewhen U is the complement of a projective variety in a projective space is a very important case.In fact the truth of Conjecture 2.20 would ensure that to bound the homogeneous arithmeticalrank from below it would be enough to work just with `etale cohomology, and not with sheafcohomology. Since usually one is interested in computing the number of (set-theoretically)defining equations of a projective variety in the projective space, in some sense the most inter-esting case of Conjecture 2.20 is when U = P n \ X for some projective variety X . From thispoint of view, one can look at Theorem 2.21 in the following way: In order to give a lowerbound for the minimal number of hypersurfaces of P n C cutting out set-theoretically a smoothprojective variety X ⊆ P n C , it is better to compute ´ecd ( P n C \ X ) than cd ( P n C \ X ) . Unfortunately, Lyubeznik informed the author of this paper by a personal communicationthat he found a counterexample, yet unpublished, to Conjecture 2.20 when the characteristic ofthe base field is positive: his counterexample consists in a scheme U which is the complementin P n of a reducible projective scheme. In this section finally we present the defining equations of the varieties described in the intro-duction. The main tools we use come from ASL theory.11 .1 Notation
We want to fix some notation that we will use throughout this section. Let k be a field ofarbitrary characteristic.We recall that the Segre product of two finitely generated N -graded k -algebra A and B isdefined as A ♯ B = M n ∈ N A n ⊗ k B n . This is a N -graded k -algebra and it is a direct summand of the tensor product A ⊗ k B .Fix n , m integers greater than or equal to 1. Then X ⊆ P n and Y ⊆ P m will always denotetwo projective schemes defined respectively by the standard graded ideals a ⊆ R = k [ x , . . . , x n ] and b ⊆ S = k [ y , . . . , y m ] .Consider the Segre product Z = X × Y and set A = R / a and B = S / b . Then we have that Z ∼ = Proj ( A ♯ B ) . Moreover, if W : = k [ x i y j : i = , . . . , n ; j = , . . . , m ] ⊆ k [ x , . . . , x n , y , . . . , y m ] = R ⊗ k S , then A ♯ B = W / I with I ⊆ W an homogeneous ideal. Assuming that a = ( f , . . . , f r ) and b = ( g , . . . , g s ) with deg f i = d i and deg g j = e j , it is easy to see that I is generated by thefollowing polynomials:1. M · f i where M varies among the monomials in S d i for every i = , . . . , r ;2. g j · N where N varies among the monomial in R e j for every j = , . . . , s .Now we present A ♯ B as a quotient of a polynomial ring. So consider P = k [ z i j : i = , . . . , n : j = , . . . , m ] and the k -algebra homomorphism f : P −→ A ♯ B defined as f = f ′ ◦ p where f ′ : P −→ W maps z i j to x i y j and p : W −→ A ♯ B ∼ = W / I is the projection. Therefore set I = Ker f . With this notation, then, we have V ∼ = Proj ( P / I ) ⊆ P N , N = nm + n + m Now we describe a system of generators which we will use in this section. For all monomials M ∈ S d i (where i = , . . . , r ), choose a polynomial ( f i ) M ∈ P such that f ′ (( f i ) M ) = M · f i . in thesame way choose a polynomial ( g j ) N ∈ P for all monomials N ∈ R e j and j = , . . . , s . Then itis easy to show that I = I ( Z ) + J where1. I ( Z ) denotes the ideal generated by the 2-minors of the matrix Z = ( z i j ) ;2. J = (( f i ) M , ( g j ) N : for all i = , . . . , r and for all monomials M ∈ S d i , forall j = , . . . , s and for all monomials N ∈ R e j ) . Our purpose is to exhibit a minimal set of defining equations (up to radical) for I in P , and so tocompute the arithmetical rank of I . We are able to solve this problem for certain ideals a and b .We need the following remark to work with algebraically closed fields and to use the Null-stellensatz: Remark 3.1.
Let H be a k -algebra and h ⊆ H an ideal. Set H ¯ k = H ⊗ k ¯ k and h ¯ k = h H ¯ k ⊆ H ¯ k ,where ¯ k denotes the algebraic closure of k . Because ¯ k is faithfully flat over k , if h , . . . , h t ∈ h are such that p h ¯ k = p ( h , . . . , h t ) H ¯ k , then √ h = p ( h , . . . , h t ) .12n the following remark we make use of an argument that we will be used several times lateron. Remark 3.2.
Actually the described generators of I are too much: for instance for a polynomial f i of the starting ideal we have to consider all the polynomials ( f i ) M with M varying in S d i .These are (cid:0) m + d i m (cid:1) polynomials! Anyway, up to radical, it is enough to choose m + f i and n + g j .For every i = , . . . , r and l = , . . . , m , set M = y d i l . A possible choice for ( f i ) M is ( f i ) l : = f i ( z l , . . . , z nl ) ∈ P . In the same manner for every j = , . . . , s and k = , . . . , n we define ( g j ) k = g j ( z k , . . . , z km ) ∈ P . We claim that √ I = p I ( Z ) + J ′ where J ′ is the ideal generated by the ( f i ) l ’s and the ( g j ) k ’s.We can assume that k is algebraically closed by Remark 3.1. So, denoting by Z ( L ) thezeroes locus of an ideal L , it is enough to prove that Z ( I ) = Z ( I ( Z ) + J ′ ) by Nullstellensatz.So pick P = [ P , P , . . . , P n , P , . . . , P n , . . . , P m , . . . , P nm ] ∈ Z ( I ( Z ) + J ) . We can write P =[ P , . . . , P m ] , where P h = [ P h , . . . , P nh ] is [ , , . . . , ] or a point of P n . Since P ∈ Z ( I ( Z )) itfollows that the non-zero points among the P h ’s are equal as points of P n . Moreover, if P l isa non-zero point, ( f i ) l ( P ) = i = , . . . , r means that P l ∈ X : then from the discussionabove trivially ( f i ) M ( P ) = i , M and any choice of ( f i ) M . By symmetry one can provethat also all the ( g j ) N ’s vanish on P , so we conclude. Remark 3.3.
Assume that X = V + ( F ) ⊆ P nk is a projective hypersurface ( F = f ), m = Y = P . We already know from a general theorem of Eisenbud and Evans (see [EE, Theorem2]) that ara ( I ) ≤ ara h ( I ) ≤ N = n + . In this case we can find an explicit set of polynomials which generate I up to radical. In fact,from a theorem of Bruns and Schw´anzl (see [BS, Theorem 2]), we know thatara ( I ( Z )) = ara h ( I ( Z )) = n − I ( Z ) up to radical: using the notationof [BrVe], set [ i , j ] = z i z j − z j z i for 0 ≤ i < j ≤ n . Then I ( Z ) = r ( (cid:229) i + j = k [ i , j ] : k = , . . . , n − ) (see [BrVe, Lemma 5.9]).By Remark 3.2 we have only to add to these generators F = ( f ) and F = ( f ) (with thenotation of Remark 3.2), and so we find 2 n + I upto radical. Theorem 3.4.
Let X = V + ( F ) ⊆ P n be a hypersurface such that there exists a line L ⊆ P n thatmeets X only at a point P, and let I be the ideal defining the Segre product X × P ⊆ P n + .Then ara h ( I ) ≤ n roof. By a change of coordinates we can assume that L = V + (( x , . . . , x n − )) . The set W = { [ i , j ] : i < j , i + j ≤ n − } is an ideal of the poset of the minors of the matrix Z = ( z i j ) (i.e. if [ i , j ] ∈ W , h ≤ i and h < k ≤ j then [ h , k ] ∈ W ), so by [BrVe, Lemma 5.9]ara ( W R ) ≤ rank ( W ) = n − . We want to prove that I = √ J where J = W R + ( F , F ) (with the notation of remarks 3.2, 3.3).To this purpose we may assume that k is algebraically closed by Remark 3.1, and we will provethe equivalent condition, by Nullstellensatz and Remark 3.2, that Z ( I ( Z ) + ( F , F )) = Z ( J ) .Let be Q = [ Q , Q ] = [ Q , . . . , Q n , Q , . . . , Q n ] ∈ Z ( J ) . If Q = Q = Q ∈ Z ( I ( Z )) , so we assume that Q , Q are points of P n . First suppose Q i j = j ≤ n − i = ,
1: for any h = k different from j , [ h , j ] (or [ j , h ] ) and [ k , j ] (or [ j , k ] ) are elements of W ,so since Q ∈ Z ( J ) we easily obtain the relations Q h Q k = Q h Q k , from which Q ∈ Z ( I ( Z )) .We can therefore assume that Q i j = j < n − , i = ,
1. But then Q and Q belong to L ∩ X , so Q = Q = P , so Q ∈ Z ( I ( Z )) . Corollary 3.5.
Let X ⊆ P be a smooth curve of degree d ≥ such that there exists a line L ⊆ P that meets X only at a point P, and let I be the ideal defining the Segre product X × P ⊆ P .Then ara h ( I ) = . Moreover, if k has characteristic , then ara ( I ) = ara h ( I ) = .Proof. Theorem 3.4 implies that ara h ( I ) ≤
4. For the lower bound first assume that k is alge-braically closed. Since X has positive genus, Proposition 2.17 implies that ´ecd ( P \ ( X × P )) ≥
8. Thus equation (10) of Remark 2.11 implies that ara h ( I ) ≥
4. If k is not algebraically closed,it is obvious that ara h ( I ) ≥ ara h ( I ( P ⊗ k ¯ k )) , so we have proved the first statement.If char ( k ) = ( P , I ) ≥
4, so ara ( I ) ≥ Remark 3.6.
In light of Theorem 3.4 and Corollary 3.5, it is natural to define the following set.For every natural numbers n , d ≥ V n − d = { X ⊆ P n : X smooth, dim X = n − , deg X = d , ∃ P ∈ X as in 3.4 } / PGL n ( k ) Notice that all hypersurfaces in V n − d can be represented, by a change of coordinate, by V + ( F ) with F = x dn − + (cid:229) n − i = x i G i ( x , . . . , x n ) , where the G i ’s are homogeneous polynomials of degree d − n =
2, and for simplicity we will write V d instead of V d . So ourquestion is: How many smooth projective plane curves of degree d do belong to V d ? First we list some plane projective curves belonging to V d .1. Every smooth elliptic curve belongs to V : in fact every smooth curve of degree greaterthan or equal to 3 has an ordinary flex, and an elliptic curve meets a line at most to3 points. So we recovered [SW, Theorem 1.1] as a consequence of Corollary 3.5 (thegenerators up to radical are different).2. Obviously, every smooth conic belongs to V too.3. Every Fermat’s curve of degree d , i.e. a projective curve C = V + ( F ) ⊆ P where F = x do + x d + x d , belongs to V d : in fact one has only to consider the line V + ( x + a x ) where a d = − [ a , , ] ∈ C , so we recovered also [So, Theorem 2.8] (the generators areagain different). 14n their paper [CD, Theorem A], Casnati and Del Centina compute the dimension of the loci V d , a , a = ,
2, of all the smooth plane curves of degree d with exactly a points as in Theorem3.4 (if these points are non singular, as in this case, they are called d -flexes), and showed that V d , a are irreducible rational locally closed subvarieties of the moduli space M g of curve ofgenus g = (cid:0) d − (cid:1) . The dimension of V d , a isdim ( V d , a ) = (cid:18) d + − a (cid:19) − + a . Moreover, it is not difficult to show that V d , is an open Zariski subset of V d , (see [CD, Lemma2.1.2]), and so dim ( V d ) = (cid:18) d + (cid:19) − . The locus H d of all smooth plane curves of degree d up to isomorphism is an open Zariski sub-set of P ( d + ) C modulo the group PGL ( C ) , so its dimension is (cid:0) d + (cid:1) −
9. Then the codimensionof V d in H d , provided d ≥
3, is d − C in the hypersurface V of H , Corolloary 3.5 impliesthat C × P ⊆ P can be defined by exactly four equations. However it remains an open problemto compute the arithmetical rank of Y × P ⊆ P for any quartic Y ⊆ P .In the general case ( n ≥ Lemma 3.7.
Let X ⊆ P n be a smooth hypersurface of degree d. If d ≤ n − , or if d ≤ n − and X is generic, then X ∈ V n − d .Proof. First we prove the following claim:a). If X ⊆ P n is a smooth hypersurface of degree d ≤ n − X ∈ V n − d .We denote by Grass ( , n ) the Grassmannian of lines of P n . Consider the projective variety W n = { ( P , L ) ∈ P n × Grass ( , n ) : P ∈ L } . It turns out that this is an irreducible variety ofdimension 2 n −
1. Now set T n , d = { (( P , L ) , F ) ∈ W n × L n , d : i ( L , V + ( F ) , P ) ≥ d } , where by L n , d we denote the projective space of all the homogeneous polynomials of degree d of K [ x , . . . , x n ] , and by i ( L , V + ( F ) ; P ) the multiplicity intersection of L and V + ( F ) at P (if L ⊆ V + ( F ) then i ( L , V + ( F ) ; P ) = + ¥ ).Assume that P = [ , , . . . , ] and that L is given by the equation x = x = . . . = x n = F ∈ L n , d the condition ( P , L , F ) ∈ T n , d is equivalentto the fact that the coefficients of x d , x d − x , . . . , x x d − in F are 0. This implies that T n , d is aclosed subset of P n × Grass ( , n ) × L n , d : thus T n , d is a projective scheme over k .Consider the restriction of the first projection p : T n , d −→ W n . Clearly p is surjective;moreover it follows by the above discussion that all the fibers of p are projective subspaces of L n , d of dimension dim ( L n , d ) − d . Therefore T n , d is an irreducible projective variety of dimension2 n − + dim ( L n , d ) − d .Now consider the restriction of the second projection p : T n , d −→ L n , d . Clearly V + ( F ) ∈ V n − d provided it is smooth, it does not contain any line and it belongs to p ( T n , d ) . So to15onclude it is enough to check the surjectivity of p whenever d ≤ n −
1. To this aim, since both T n , d and L n , d are projective, it is enough to show that for a general F ∈ p ( T n , d ) , the dimensionof the fiber p − ( F ) is exactly2 n − − d . On the other hand it is clear that the codimension of p ( T n , d ) in T n , d is at least d − n + d ≥ n . We proceed by induction on n (for n = d ≤ n −
3. Let F be a general form of p ( T n , d ) , and set r = dim ( p − ( F )) . By contradiction assume that r > n − − d . Consider a general hyperplanesection of V + ( F ) , and set F ′ the polynomial defining it. Obviously any element of p ( T n − , d ) comes from p ( T n , d ) in this way, so F ′ is a generic form of p ( T n , d ) . The condition for a lineto belong to a hyperplane is of codimension 2, so the dimension of the fiber of F ′ is at least r −
2. Since F ′ is a polynomial of K [ x , . . . , x n − ] of degree d ≤ ( n − ) −
1, we can apply aninduction getting r − ≤ n − − d , so that r ≤ n − − d , which is a contradiction.We end with the case in which d = n − d = n − F and r beas before, and suppose by contradiction that r ≥
1. This implies that there exists a hypersurface H ⊆ Grass ( n − , n ) such that for any general H ∈ H the polynomial defining V + ( F ) ∩ H belongs to p ( T n − , d ) . This implies that the codimension of p ( T n − , d ) in T n − , d is less than orequal to 1, whereas we know that this is at least 2.So a) holds true. Now we prove the lemma by induction on n (if n = d ≤ n −
3, then we cut X by a generic hyperplane H . It turns out (using Bertini’s theorem)that X ∩ H ⊆ P n − is the generic smooth hypersurface of degree d ≤ ( n − ) −
1, so by inductionthere exist a line L ⊆ H and a point P ∈ P n such that ( X ∩ H ) ∩ L = { P } . So we conclude that X ∈ V n − d .It is known that the generic hypersurface of degree d ≥ n − d = n − d = n − Corollary 3.8.
Let X ⊆ P n be a smooth hypersurface of degree d, and let I be the ideal definingthe Segre product X × P ⊆ P n + . If d ≤ n − , or if d ≤ n − and X is generic, then ara h ( I ) ≤ nProof. Just combine the above lemma with Theorem 3.4.Putting some stronger assumptions on the hypersurfaces we can even compute the arith-metical rank of the ideal defining their Segre product with P (and not just give an upper boundas in Theorem 3.4). Theorem 3.9.
Let X = V + ( F ) ⊆ P n be such that, F = x dn + (cid:229) n − i = x i G i ( x , . . . , x n ) (G i homoge-neous polynomials of degree d − ), and let I be the ideal defining the Segre product X × P ⊆ P n + . Then ara h ( I ) ≤ n − . Moreover, if X is smooth, then ara h ( I ) = n − . Furthermore, if k has characteristic 0, then ara ( I ) = ara h ( I ) = n − .Proof. We can assume that k is algebraically closed. If X is smooth, by Proposition 2.16´ecd ( P n + \ X ) ≥ n −
1, and equation 10 yields ara h ( I ) ≥ n −
1. If char ( k ) = ( I ) ≥ n −
1. 16ow we prove that the upper bound holds. Consider the set W = { [ i , j ] : i < j , i + j ≤ n − } .As in the proof of Theorem 3.4, we haveara ( W R ) ≤ rank ( W ) = n − . Now the proof is completely analogue to that of Theorem 3.4.
Remark 3.10.
Notice that, if n ≥
4, the generic hypersurface of P n defined by the form F = x dn + (cid:229) n − i = x i G i ( x , . . . , x n ) is smooth (whereas if n ≤ d ≥ ( k ) . Theorem 3.11.
Assume char ( k ) = . Let C = V + ( F ) be a conic of P , and let I be the idealdefining the Segre product X = C × P m ⊆ P m + . Then ara h ( I ) = m . In particular X is a set-theoretic complete intersection if and only if m = . Moreover, if char ( k ) = , then ara ( I ) = ara h ( I ) = m.Proof. First we want to give 3 m homogeneous polynomials of S = k [ z i j : i = , , , j = , . . . , m ] which define I up to radical.For i = , . . . , m choose F i as in Remark 3.3. Then, for all 0 ≤ j < i ≤ m , set F i j = (cid:229) k = ¶ F ¶ x k ( z i , z i , z i ) z jk . Finally we set G h = (cid:229) i + j = h F i j for all h = , . . . , m −
1. We claim that I = √ J , where J = ( F i , G j : i = , . . . , m , j = , . . . , m − ) . The inclusion J ⊆ I follows from the Euler’s formula, since char ( k ) = k algebraically closed by Remark 3.1, so we have to prove that I ⊆ √ J , i. e.,by the Nullstellensatz, that Z ( J ) ⊆ Z ( I ) . Pick P ∈ Z ( J ) , and write P as P = [ P , P , . . . , P m ] where P j = [ P j , P j , P j ] . Since F i ( P ) =
0, for every i = , . . . , m P i = P i ∈ C . So we haveto prove that the P i ’s that are non zero are equal as points of P .By contradiction, let i be the minimum integer such that P i = k such that P k = P i = P k as points of P , and set j the least among these k (so i < j ). Set h = i + j .We claim that P k = P l provided that k + l = h , k < l , k = i , P k = P l = l < j , then P i = P l by the choice of j . But for the same reason also P k = P i , so P k = P l .If l > j , then k < i , so it follows that P k = P l by the choice of i . So F lk ( P ) =
0, because P k belongs to the tangent of C in P l (being P l = P k ). Then G h ( P ) = F ji ( P ) , and so, since P ∈ Z ( J ) , F ji ( P ) =
0: this means that P i belongs to the tangent line of C in P j , which is possible, being C a conic, only if P i = P j , a contradiction.For the lower bound, we can assume that k is algebraically closed as in the proof of Corollary3.5. By Proposition 2.16 ´ecd ( P n + \ X ) ≥ n −
1, and equation (10) yields ara h ( I ) ≥ n −
1. Ifchar ( k ) = ( I ) ≥ n − emark 3.12. B˘adescu and Valla computed recently in [BaVa], independently from this work,the arithmetical rank of the ideal defining any rational normal scroll. Since the Segre productof a conic with P m is a rational normal scroll, Theorem 3.11 is a particular case of their result.We end the paper with a proposition that yields a natural question. Proposition 3.13.
Let X = V + ( F ) ⊆ P n be a hypersurface smooth over a field of characteristic and let I ⊆ P = k [ z , . . . , z N ] be the ideal defining Z = X × P m ⊆ P N (any embedding), withm ≥ . Then cd ( P , I ) = (cid:26) N − if n = and deg ( F ) ≥ N − otherwiseProof. By Remark 2.2 we can assume k = C . Using equation (5) we have b ( Z ) = , b ( Z ) = b ( X ) and b ( Z ) = b ( X ) + ≥ . If n =
2, notice that b ( X ) = ( F ) ≥
3. In fact, by equation (6), b ( X ) = h ( X ) + h ( X ) = h ( X ) (the last equality comes from Serre’s duality). But h ( X ) is the geometric genus of X , thereforeit is different from 0 if and only if deg ( F ) ≥
3. So if n = n > b ( X ) = Question 3.14.
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