Sheng-Li Tan

On the invariants of base changes of pencils of curves, I

The main purpose of this paper is to prove the nonnegativity of the basic invariants of base changes of a surface fibration, which is conjectured by Xiao Gang. For this purpose we obtain some new inequalities between the invariants of the singularities ofzd=f(x, y).

ON THE SLOPES OF THE MODULI SPACES OF CURVES

Let Mg be the moduli space of smooth curves of genus g and Mg the moduli space of stable curves of genus g. Then Mg = MgU A, where A = YJg£ ] A*> Ao is the closure of the locus of genus g — 1 curves with one double point, and A* (i ^ 0) is the the closure of the locus of the stable curves of type (i,g — i). Let A be the class of the Hodge line bundle on A4S, let Si be the class of Aj if i ^ 1, and let Si be half of the class of Ai . Let S = So H h 4 [11], and on the moduli functor, the canonical class is 13A — 2S, the only difference is the coefficient of Si (see [11, 9] for the details). This class is called effective if n(aX — bS) is effective for sufficiently large and sufficiently divisible n. Harris and Morrison [10] define the slope of the moduli space as

On the invariants of base changes of pencils of curves, II

In this part of the series, we shall investigate Deligne-Mumford semistable reductions from the point of view of numerical invariants. As an application, we obtain two numerical criterions for a base change to be stabilizing, and for a fibration to be isotrivial. We also obtain a canonical class inequality for any fibration. Some other applications are presented. Most of the results of this paper have arithmetical analogues. This paper will appear in Math. Z.

Integral closure of a cubic extension and applications

In this paper, we compute the integral closure of a cubic extension over a Noetherian unique factorization domain. We also present some applications to triple coverings and to rank two reflexive sheaves over an algebraic variety.

The determination of integral closures and geometric applications

Abstract We express explicitly the integral closures of some ring extensions; this is done for all Bring–Jerrard extensions of any degree as well as for all general extensions of degree ⩽5; so far such an explicit expression is known only for degree ⩽3 extensions. As a geometric application, we present explicitly the structure sheaf of every Bring–Jerrard covering space in terms of coefficients of the equation defining the covering; in particular, we show that a degree-3 morphism π:Y→X is quasi-etale if and only if c 1 (π ∗ O Y ) is trivial (details in Theorem 5.3). We also try to get a geometric Galoisness criterion for an arbitrary degree-n finite morphism; this is successfully done when n=3 and less satisfactorily done when n=5.

Lojasiewicz inequality for weighted homogeneous polynomial with isolated singularity

Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.

Proceedings of the international conference on Complex Geometry and Related Fields

Representations of two-parameter quantum orthogonal and symplectic groups by N. Bergeron, Y. Gao, and N. Hu Optimal control of the Liouville equation by R. W. Brockett Real structures on torus bundles and their deformations by F. Catanese and P. Frediani Cubic equations of rational triple points of dimension two by Z. Chen, R. Du, S.-L. Tan, and F. Yu Multi-parameter cells of finite Coxeter groups by J. Du and H. Rui On some rigidity problems in Cauchy-Riemann analysis by X. Huang and S. Ji From CR geometry to algebraic geometry and combinatorial geometry by L. Jia, H. S. Luk, and S. S.-T. Yau Periods of automorphic forms by D. Jiang Some problems related to hamiltonian line graphs by H.-J. Lai and Y. Shao Composition operators and isometries on holomorphic function spaces over domains in

A note on the Cayley-Bacharach property for vector bundles

\mathbb{C}^n

Optimal control of the Liouville equation

by S.-Y. Li Localization and string duality by K. Liu Polyhedral and geometric convergence of Kleinian groups by Z. Long, X. Wang, and Y. Wang Rigidity problems on compact quotients of bounded symmetric domains by N. Mok The second main theorem with hypersurfaces over function fields by M. Ru Presentations for finite complex reflection groups by J.-Y. Shi Representations of finite Lie algebras and geometry of reductive Lie algebras by B. Shu Perspectives on geometric analysis by S.-T. Yau Automorphisms of K3 surfaces by D.-Q. Zhang Vector bundles on certain surfaces without divisors by L. Zhao, X. Zhou, and Q. Li.

Surfaces whose canonical maps are of odd degrees

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 ys|Z′ ys 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 ‖ x x 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 xφ1 xφ2 xψ 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