Frank-Olaf Schreyer

Complex Analysis and Algebraic Geometry: A Volume in Memory of Michael Schneider

Lucian Badescu, Mauro C. Beltrametti, Paltin Ionescu, almost-lines and quasi-lines on projective manifolds Daniel Barlet, Jon Magnusson, Transfert de metrique Wolf P. Barth, on the classification of K3 surfaces with nine cusps Arnaud Beauville, complex manifolds with split tangent bundle Mauro C. Beltrametti, Alan Howard, Michael Schneider, Andrew J. Sommese, projections from subvarieties Indranil Biswas, Georg Schumacher, generalized Petersson-Weil metric on the Douady space of embedded manifolds Fabrizio Catanese, Roberto Pignatelli, on simply connected Godeaux surfaces Ciro Ciliberto, Angelo Felice Lopez, Rick Miranda, on the Wahl map of plane nodal curves Lawrence Ein, Bo Ilic, Robert Lazarsfeld, a remark on projective embeddings of varieties with non-negative cotangent bundles David Garber, Mina Teicher, the fundamental groups structure of the complement of some configurations of real line arrangements Peter Heinzner, Alan T. Huckleberry, Kahlerian structures on symplectic reductions Klaus Hulek, Nef divisors on moduli spaces of Abelian varieties Klaus Hulek, Kristian Ranestad, Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds Janos Kollar, real algebraic threefolds IV. Del Pezzo fibrations Christian Okonek, Andrei Teleman, Seiberg-Witten invariants for 4-manifolds with b+ = 0 Jeroen Spandaw, a geometric proof of Ax Theorem Sheng-Li Tan, Eckart Viehweg, a note on Cayley-Bacharach property for vector bundles Thomas Peternell, the scientific work of Michael Schneider Ulf Persson, Michael Schneider - an alpine vita.

A note on the Cayley-Bacharach property for vector bundles

We study the Cayley-Bacherach property on smooth complex projective varieties for zero-dimensional subschemes, defined as the zero set of a global section of a rank n vector bundle, and for codimension 2 subschemes, defined by global sections of rank 2 vector bundles. 1991 Mathematics Subject Classification: 14J60, 14F05, 14M06, 14N99. The main purpose of this note is to present and to generalize results from [17] and to use them to study properties and the construction of vector bundles on smooth complex projective varieties X of dimension n ≥ 2. In [17], the first author proved that the Cayley-Bacharach property of a zerodimensional complete intersection in X is equivalent to the k-very ampleness of some adjoint linear systems. In this paper, we show that the result remains true for the zero-dimensional subscheme defined by the zero set of a global section of a rank n vector bundle (Theorem 7), generalizing a theorem of Griffiths and Harris [8], p.677. Due to the Bogomolov inequality for rank 2 semistable vector bundles [4] [12], we can establish the Cayley-Bacharach theorem for codimension 2 subschemes defined by global sections of rank 2 vector bundles (Theorem 8 and Corollary 9). This result can be used to reprove Paoletti’s theorem [14] [13], a generalization of the classical theorem of Halphen. As an application, we give an explicit construction of rank 2 vector bundles from codimension 2 subschemes (Theorem 10). Throughout this paper we use the notion “k points” for any zero-dimensional subscheme of length k, not requiring the points to be distinct. The degree of an object is defined with respect to an fixed ample divisor A on X, hence the degree of a codimension r subscheme Y of X is defined by deg Y = An−rY , although A is not mentioned in the statements. ∗This work is supported by the DFG Forschergruppe “Arithmetik und Geometrie”. The first author is also supported by the NSF for Outstanding Youths. 2 S.-L. Tan and E. Viehweg 1. An Exact Sequence Let X be a smooth projective variety over C of dimension n ≥ 2, and let Z be a subscheme of X of pure codimension r ≥ 2. Given a subscheme Z ′ ⊂ Z, the “complement” Z ′′ of Z ′ in Z is the canonical closed subscheme Z ′′ ⊂ Z with sheaf of ideals IZ′′ = [IZ : IZ′ ], i.e., for any open set U ⊂ X, we define IZ′′(U) := {g ∈ OX(U) | gIZ′(U) ⊂ IZ(U)}, or equivalently, IZ′′/IZ = HomOX (OZ′ ,OZ). The second description implies that Z ′′ = Z if the support of Z ′ does not contain some of the irreducible components of Z. Moreover, if Z is reduced, then Z ′′ is the closure of Z −Z ′. We call Z ′′ the residual subscheme of Z ′ in Z and denote it by Z ′′ = Z − Z ′. Let E be a vector bundle on X of rank r ≥ 2, let s be a global section of E and let Z = Z(s) ⊂ X be its zero scheme. As above we will assume that Z is of pure codimension r, hence it is a local complete intersection. For a divisor L and a subscheme ∆ ⊂ Z(s), we want to study hypersurfaces F in X satisfying the equations { ∆ = Z(s)− Z(s)F, L ≡ detE − F. (∗) Given ∆ and L we will call (E, s, F ) a solution of (∗) if Z(s) is of pure codimension r = rank(E) and if the equation (∗) holds true. Here and throughout this note Z(s)F denotes the intersection subscheme of Z(s) and a hypersurface (or effective divisor) F in X. If a hypersurface F satisfies the first equation in (∗) we will say that F does not pass through ∆. In a similar way, if Z ′ is a subscheme of F , we will say that F passes through Z ′. If Z(s) is a reduced subscheme of X then F satisfies the first equation in (∗), if ∆ is the union of all irreducible components of Z(s) which are not contained in F . Theorem 1. Let E be a vector bundle on X of rank r ≥ 2, and let s be a section whose zero subscheme Z = Z(s) is of pure codimension r. Let Z ′′ ⊂ Z ′ are two codimension r subschemes of Z and let L be a divisor. Then there exists a complex of vector spaces 0 −→ H(IZ−Z′′( detE − L)) α −→H(IZ−Z′(detE − L)) μ −→ H(IZ′(KX + L)) β −→ H(IZ′′(KX + L)) −→ 0, Cayley-Bacharach Property for Vector bundles 3 exact except at H(IZ′(KX+L)). If E is sufficiently ample, then the complex is exact everywhere. Remark. The condition “E is sufficiently ample” we used in the theorem stands for the following vanishing conditions: Hj(X,∧iE∨(detE − L)) = 0, for i, j = 1, · · · , r − 1. (1) If X = P and E splits (the hypersurface case), then (1) is always true. In general (1) can be enforced by replacing E by E ⊗H, for a sufficiently ample line bundle H (cf. Lemma 4 and the end of the proof of Theorem 1). The connecting map μ is not “natural”, but there is natural map to the dual of kerβ. Throughout the proof of Theorem 1, F will denote an effective divisor on X with F ≡ detE − L. We consider the Koszul complex of (E, s): 0 −→ Er−1 −→ · · · −→ E0 s −→IZ −→ 0, where E0 = E∨, Ei = ∧E0, and where s is the dual map of O → E∨ 0 given by the global section s of E∨ 0 . Because Z = Z(s) is a local complete intersection, the Koszul complex is exact (see [9], p.245). We split the Koszul complex as follows: 0 −→ F1 −→ E0 s −→ IZ −→ 0, 0 −→ F2 −→ E1 −→ F1 −→ 0, .. .. (2) 0 −→ Fr−1 s −→ Er−2 −→ Fr−2 −→ 0, where Fr−1 ∼= Er−1 ∼= detE∨. Lemma 2. Assume that Z ′ is a subscheme of Z. If (1) holds, then H(IZ′(KX + L))∨ ∼= Ext(IZ′ ,F1(F )). Proof. By Serre duality ([9], Theorem 7.6) one has an isomorphism H(IZ′(KX + L))∨ ∼= Ext(IZ′ ,O(−L)). On the other hand, (1) implies that Ext(OX , Ei(F )) ∼= H(Ei(F )) = 0, for i ≤ r − 2, 1 ≤ j ≤ r − 1. (3) From (3) and from the exact sequence 0 −→ IZ′ −→ OX −→ OZ′ −→ 0, (4) we obtain easily that Ext(IZ′ , Ei(F )) ∼= Ext(OZ′ , Ei(F )) ∼= H(OZ′(E i (−F +KX))) = 0, 4 S.-L. Tan and E. Viehweg for i ≤ r − 2, 1 ≤ j ≤ r − 2. Considering the long exact sequences obtained from the short exact sequences in (2), we thereby have isomorphisms Ext(IZ′ ,F1(F )) ∼= Ext(IZ′ ,F2(F )) ∼= · · · ∼= Ext(IZ′ ,Fr−2(F )) and an exact sequence 0 −→ Ext(IZ′ ,Fr−2(F )) −→ Ext(IZ′ ,Fr−1(F )) τ −→ Ext(IZ′ , Er−2(F )). Since Fr−1(F ) ∼= O(−L), it remains to prove that the morphism τ is zero. Indeed, E∨ r−2 ∼= E0 ⊗ detE and by Serre duality τ is the dual morphism of H(IZ′ ⊗ E0(KX + L)) s −→ H(IZ′(KX + L)). On the other hand, from (4), we obtain a commutative diagram H(OZ′ ⊗ E0(KX + L)) −−→ H(IZ′ ⊗ E0(KX + L)) −−→ 0 uf8e6ys|Z′ uf8e6ys H(OZ′(KX + L)) −−→ H(IZ′(KX + L)) Since s is vanishing on Z ′, we find s|Z′ to be zero, which implies that the morphism s is zero as well. ut Lemma 3. Under the assumptions made in Lemma 2, there is an exact sequence H(E0(F )) = Hom(IZ′ , E0(F )) s −→H(IZ−Z′(F )) −→ Ext(IZ′ ,F1(F )) −→ 0. Proof. Applying the functor Hom(IZ′ , ·) to 0 −→ F1(F ) −→ E0(F ) s −→IZ(F ) −→ 0, we obtain the exact sequence Hom(IZ′ , E0(F )) s −→Hom(IZ′ , IZ(F )) −→ Ext(IZ′ ,F1(F )) −→ 0. Note that the 0 term on the right hand side comes from (3) if r ≥ 3, and for r = 2 from the morphism τ : Ext(IZ′ ,Fr−1(F )) −→ Ext(IZ′ , Er−2(F )) which is zero as we have seen in the proof of Lemma 2. Because Z − Z ′ is the residual subscheme of Z ′ in Z, we have (cf. [17]) Hom(IZ′ , IZ(F )) ∼= H(IZ−Z′(F )), completing the proof of Lemma 3. ut Cayley-Bacharach Property for Vector bundles 5 Proof of Theorem 1 for E sufficiently ample. By Lemma 2 and Lemma 3 for Z ′ and Z ′′, we obtain a commutative diagram 0 −−→ Im s −−→ H(IZ−Z′(F )) μZ′ −−→ H(IZ′(KX + L))∨ −−→ 0 ‖ xuf8e6 xuf8e6 0 −−→ Im s −−→ H(IZ−Z′′(F )) −−−→ μZ′′ H(IZ′′(KX + L))∨ −−→ 0 (5) Note that the middle and right vertical morphisms are injective and by the Five Lemma we can see that they have the same cokernel Q, hence 0 −→ H(IZ−Z′′(F ) −→ H(IZ−Z′(F )) −→ Q −→ 0, and 0 −→ Q∨ −→ H(IZ′(KX + L)) −→ H(IZ′′(KX + L)) −→ 0. Choosing any isomorphism Q ∼= Q∨ one obtains Theorem 1 from the two exact sequences above. ut For the general case we will replace the vector bundle E by E ⊗H, for some sufficiently ample line bundle H. Lemma 4. Assume that (E, s, F ) is a solution of (∗) for fixed L and ∆. Let H be a sufficiently ample line bundle and M ∈ H(E ⊗ E∨ ⊗H) a sufficiently general section, viewed as a morphism M : E → E ⊗H. Let Ẽ = E ⊗H, s̃ = sM, F̃ = F + Z(detM). Then (Ẽ, s̃, F̃ ) is also a solution of (∗) for L and ∆. Proof. We can assume that the divisor of detM does not contain any component of Z(s). Let ∆̃ = Z(s̃)− Z(s̃)F̃ be the new residual subscheme. We only need to prove that ∆̃ = ∆, i.e., Ie ∆ = I∆. Indeed, by definition, it is clear that I∆ ⊂ Ie ∆. Conversely, Ie ∆ consists of the local sections g̃ such that g̃f detM vanishes on Z(s̃), where f is the local defining equation of F . Hence it also vanishes on Z(s). Because detM does not vanish on any component of Z(s), this implies that g̃f vanishes on Z(s). Now we know that g̃ is contained in I∆. So Ie ∆ ⊂ I∆. ut Proof of Theorem 1 for arbitrary E. Keeping the notations and assumptions of Lemma 4 we have a diagram 0 −−→ H0(Ie Z−Z′′(F̃ )) −−→ H0(Ie Z−Z′(F̃ )) −−→ Q̃ −−→ 0 xuf8e6φ1 xuf8e6φ2 xuf8e6ψ 0 −−→ H(IZ−Z′′(F )) −−→ H(IZ−Z′(F )) −−→ Q −−→ 0 6 S.-L. Tan and E. Viehweg where Q = cokerα, and where φ1 and φ2 are defined as the multiplication by detM . In particular φ1 and φ2 are injective. For H sufficiently ample, Theorem 1 holds true for Ẽ, and Q̃ = ker β. Thus we only need to prove that ψ is injective. By the Five Lemma, it is enough to prove that the induced natural map cokerφ1 → cokerφ2 is injective. Indeed, let G ≡ F̃ represent an element of cokerφ1, then G passes through Z̃ − Z ′′. If its image in cokerφ2 is zero, i.e., if G = G′ + detM and G′ passes through Z − Z ′, we need to prove that G is also zero in cokerφ1, i.e., G′ passes through Z − Z ′′. This is obvious because detM does not pass through Z − Z ′′, but G does. ut 2. Solutions of the Equation (∗) Theorem 5. Let ∆ be a subscheme of X of pure codimension r and let L be a divisor on X. Then the following conditions are equivalent. 1) (∗) has a solution (E, s, F ) for ∆ and L, i.e., there are a hypersurface F , a rank r vector bundle E and a nonzero global section s of E whose zero set Z = Z(s) is an n − r dimensional subscheme such that (∗) holds, so ∆ is th