On Effective Non-vanishing of Weil Divisors on Algebraic Surfaces

Weil divisors on rational normal scrolls

The aim of this paper is to study Weil divisors on a singular rational normal scroll X. In particular the author describes explicitly the group of divisorial sheaves associated to Weil divisors on X, via the direct image of the Picard group of the canonical resolution of X, which is very well known. The analysis naturally splits in two cases, depending on the codimension of the vertex of X (>2 or =2). When the codimension of the vertex is >2, then the two groups are isomorphic and there is a natural intersection form in X inherited from the one in the canonical resolution. When the codimension…

Toroidal b-divisors and Monge-Ampère measures

We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of smooth and complete toroidal embeddings. As a key ingredient we show the existence of a limit measure, supported on the weakly embedded rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef toroidal Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can …

通过 Cartier 除子的表示有效计算局部 Weil 函数

Effective calculation of local Weil functions via presentations of Cartier divisors

本文从 Cartier 除子表示出发，结合 Bombieri–Gubler 方法与既往工作，建立计算射影簇上局部 Weil 函数的理论有效算法。

Abelian Fourfolds of Weil type and certain K3 Double Planes

Double planes branched in 6 lines give a famous example of K3 surfaces. Their moduli are well understood and related to abelian fourfolds of Weil type. We compare these two moduli interpretations and in particular divisors on the moduli spaces. On the K3 side, this is achieved with the help of elliptic fibrations. We also study the Kuga-Satake correspondence on these special divisors.

Weil-Poincaré series and topology of collections of valuations on rational double points

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over …

A Hodge theoretic criterion for finite Weil--Petersson degenerations over a higher dimensional base

We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of Hodge structures and variation of mixed Hodge structures. We also give a description on the codimension 2 locus for the moduli space of Calabi--Yau threefolds. We prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices. Finally, by giving a classification of the dominant term of the candidates of the Weil--Petersson potent…

On contractible curves on normal surfaces

We characterize contractible curves on proper normal algebraic surfaces in terms of complementary Weil divisors. Using this we generalize the classical criteria of Castelnuovo and Artin. As application we derive a finiteness result on homogeneous spectra defined by Weil divisors on surfaces.

The volume of an isolated singularity

We introduce a notion of volume of a normal isolated singularity that generalizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a non-invertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov's b-divisors. We define the notion…

On the ampleness and bigness of non-integral divisors

Given a Weil non-integral divisor $D$, it is natural to associate it the line bundle of its integral part $\mathcal{O}_X([D])$. In this work we study which of the classical characterizations of ample and big divisors can be extended to non-integral divisors via the corresponding line bundles.

On explicit birational geometry for polarised varieties

In this paper, we investigate the explicit birational geometry for projective $ε$-lc varieties polarised by nef and big Weil divisors. We show that if $X$ is a projective $ε$-lc variety, $H$ is a nef and big Weil divisor with $\dim\overline{φ_{H}(X)}\geq n-1$ and $L$ is an effective Weil divisor such that $|L-K_X|\neq \emptyset$ or $L-K_X$ is nef, then we can find an explicit lower bound of $\text{vol}(H)$ and prove that $|L+m^\prime H|$ is birational for $m^\prime\geq m$, where $m$ is an explicit number which depends only on $n$ and $ε$. This result can be applied to polarised Calabi-Yau vari…

Vanishing and injectivity for R-Hodge modules and R-divisors

We prove the injectivity and vanishing theorem for R-Hodge modules and R-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in \cite{Wu15}. In particular, the injectivity generalizes the fundamental injectivity of Esnault-Viehweg for normal crossing Q-divisors, while the vanishing generalizes Kawamata-Viehweg vanishing for Q-divisors. As a main application, we also deduce a Fujita-type freeness result for R-Hodge modules in the normal crossing case.

Horrocks-Mumford丛的Weil装饰

The Weil Decoration of the Horrocks-Mumford Bundle

本文研究了正常代数簇中反射性秩一sheaf与Weil除子之间的关系，并将其推广到任意秩的反射性sheaf及其所谓的Weil装饰，同时定义并研究了Horrocks-Mumford丛的一个自然推广。

Geometry of polarised varieties

In this paper we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if $X$ is a projective variety of dimension $d$ with $ε$-lc singularities for $ε>0$, and if $N$ is a nef and big Weil divisor on $X$ such that $N-K_X$ is pseudo-effective, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d,ε$. This is key to proving various other results. For example, it implies that if $N$ is a big Weil divisor (not necessaril…

What is missing in canonical models for proper normal algebraic surfaces?

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numerically equivalent to a Q-Cartier divisor. Similar results hold for arbitrary Weil divisors instead of the canonical class.

Generic vanishing in characteristic p>0 and the geometry of theta divisors

In this paper we prove a strengthening of the generic vanishing result in characteristic $p>0$ given in [HP16]. As a consequence of this result, we show that irreducible $Θ$ divisors are strongly F-regular and we prove a related result for pluri-theta divisors.

Vanishing for Hodge ideals of $\mathbb{Q}$-divisors

Mustaţă a and Popa introduce the notion of Hodge ideals for an effective $\mathbb{Q}$-divisor $D$ and prove a vanishing theorem for Hodge ideals, which generalizes Nadel vanishing for multiplier ideals. However, their proof needs an extra assumption on the existence of $\ell$-roots of the line bundle $\mathscr{O}_X(\ell D)$, which is not necessary for Nadel vanishing. In this paper, we prove that vanishing for Hodge ideals still holds even without this assumption.

Witt除子层的零化定理与一个反例

Vanishing and a counterexample for Witt divisorial sheaves

本研究完善正特征完美域上光滑射影簇的Witt除子层对偶理论，得到更简洁同构，证明了曲面nef且big除子对应Witt除子层的Ramanujam型零化定理，并给出二维下Kawamata-Viehweg型零化的反例。

除子的通用消失猜想

Generic vanishing conjecture for divisors

受Pareschi和Popa的通用消失猜想启发，本文证明了余维数为1情形的对应类比。

Vanishing theorems for linearly obstructed divisors

We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the blow-up of the space along their linear base locus. We extend the result to non-effective divisors that sit in a small region outside the effective cone. As an application, we describe linear systems of divisors in $\mathbb{P}^n$ blown-up at points in star configuration and their strict transforms via the blow-up of the linear base locus.

Decomposable theta divisors and generic vanishing

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding addition map, and show that the minimum can only be achieved if X is a theta divisor. Conjecturally, the latter happens only on Jacobians of curves and intermediate Jacobians of cubic threefolds. As an application, we prove that nondegenerate generic vanishing subschemes of indecomposable principally polarized abelian varieties are automatically reduced and irreduc…

On birational geometry of minimal threefolds with numerically trivial canonical divisors

For a minimal $3$-fold $X$ with $K_X\equiv 0$ and a nef and big Weil divisor $L$ on $X$, we investigate the birational geometry inspired by $L$. We prove that $|mL|$ and $|K_X+mL|$ give birational maps for all $m\geq 17$. The result remains true under weaker assumption that $L$ is big and has no stable base components.

A numerical ampleness criterion via Gale duality

The main object of the present paper is a numerical criterion determining when a Weil divisor of a $\Q$--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak $\Q$--Fano, $\Q$--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studi…

Kawamata-Viehweg vanishing as Kodaira vanishing for stacks

We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor $D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes integral. We then reinterpret the Kawamata--Viehweg vanishing theorem as Kodaira vanishing for stacks.

On semipositivity of sheaves of differential operators and the degree of a unipolar Q-Fano variety

We consider normal projective n-dimensional varieties X whose anticanonical divisor class -K is ample and where every Weil divisor is a rational multiple of K. The index i is the largest integer such that K/i exists as a Weil divisor. We show (i) if X has log-terminal singularities, and in addition 1-forms on the smooth part of X are holomorphic on a resolution, then (-K)^n =< (max(in,n+1))^n; (ii) if the tangent sheaf of X is semistable, then (-K)^n =<(2n)^n. The proof is based on some elementary but possibly surprising slope estimates on sheaves of differential operators on plurianticanonica…

On Spaces Associated with Invariant Divisors on Galois Covers of Riemann Surfaces and Their Applications

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.

Families of G-constellations over resolutions of quotient singularities

Let G be a finite subgroup of GL_n(C). A study is made of the ways in which resolutions of the quotient space C^n / G can parametrise G-constellations, that is, G-regular finite length sheaves. These generalise G-clusters, which are used in the McKay correspondence to construct resolutions of orbifold singularities. A complete classification theorem is achieved, in which all the natural families of G-constellations are shown to correspond to certain finite sets of G-Weil divisors, which are a special sort of rational Weil divisor, introduced in this paper. Moreover, it is shown that the number…

Coverings of rational ruled normal surfaces

In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.

Hodge ideals for Q-divisors: birational approach

We develop the theory of Hodge ideals for Q-divisors by means of log resolutions, extending our previous work on reduced hypersurfaces. We prove local (non-)triviality criteria and a global vanishing theorem, as well as other analogues of standard results from the theory of multiplier ideals, and we derive a new local vanishing theorem. The connection with the V-filtration is analyzed in a sequel.

Logarithmic vanishing theorems for effective $q$-ample divisors

Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,Ω^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities II: Vanishing Topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an ext…

关于弱丰沛除子的Kodaira--Saito消没定理

On the Kodaira--Saito vanishing theorem of weakly ample divisor

本文在光滑射影簇上，针对层上同调意义下弱丰沛的既约有效除子，引入并证明了相应的Kadaira--Saito消没定理。

复可逼近代数的上同调类

Cohomology classes of complex approximable algebras

本文证明在复数域上，与可逼近分次代数相关的无穷Weil除子具有有限上同调类。

On effective non-vanishing and base-point-freeness

Let $X$ be a complete normal variety, $B$ an effective $\mathbb{R}$-divisor on $X$, and $D$ a Cartier divisor on $X$. Assume that the pair $(X, B)$ is log terminal. We consider the problem whether $H^0(X, D) \ne 0$ and obtain some results in lower dimensions.

Bogomolov-Sommese vanishing and liftability for surface pairs in positive characteristic

We show that the Bogomolov-Sommese vanishing theorem holds for a log canonical projective surface in large characteristic unless the Iitaka dimension of the round-down of the log canonical divisor is not equal to two. As an application, we prove that a log resolution of a pair of a normal projective surface and a reduced divisor in large characteristic lifts to the ring of Witt vectors when the Iitaka dimension of the log canonical divisor is less than or equal to zero. Moreover, we give explicit and optimal bounds on the characteristic unless their Iitaka dimensions are equal to zero.

Vector Fields on Smooth Threefolds Vanishing on Complete Intersections

The existence of a vector field on a compact Kaehler manifold with nonempty zero locus and the properties of this zero locus strongly influence the geometry of the manifold. For example, J. Wahl proved that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducib…

A Nonvanishing Theorem for Q-divisors

We prove a non-vanishing theorem of the cohomology H^0 of the adjoint divisor K_X + L' where L' is the round up of a nef and big Q-divisor L.

Irreducible Theta Divisors of PPAV's are Strongly F-regular

We study the birational geometry of irregular varieties and the singularities of Theta divisors of PPAV's in positive characteristic by applying recent generic vanishing results of Hacon and Patakfalvi. In particular, we prove that irreducible Theta divisors of principally polarized abelian varieties are strongly F-regular, which extends an old result of Ein and Lazarsfeld to fields of positive characteristic. In order to prove this, we formulate a positive characteristic analogue of another result of Ein and Lazarsfeld, to the effect that the Albanese image of a smooth projective variety of m…

Vanishing thetanulls and hyperelliptic curves

Let $\mathcal{M}_{g,2}$ be the moduli space of curves of genus $g$ with a level-2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in $\mathcal{M}_{6,2}$. We prove also that for all $g\geqslant3$, each component of the hyperelliptic locus in $\mathcal{M}_{g,2}$ is a connected component of the intersection of $g-2$ thetanull divisors.

循环商曲面奇点中的普通 r-元组

Ordinary $r$--tuples in cyclic quotient surface singularities

本文刻画了循环商曲面奇点中哪些Weil除子构成抽象的r-元组曲线奇点，揭示了奇点结构与除子性质之间的对应关系。

Powers of the theta divisor and relations in the tautological ring

We show that the vanishing of the $(g+1)$-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel-Jacobi maps, the vanishing results in the tautological ring of $\mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber-Vakil, and Faber-Pandharipande. We also show that Pixton's double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem $\star$ of Graber and Vakil. Moreover, our proof provides an algorithm for expressing…

Homotopy groups of complements to ample divisors

We study the homotopy groups of complements to reducible divisors on non-singular projective varieties with ample components and isolated non normal crossings. We prove a vanishing theorem generalizing conditions for commutativity of the fundamental groups. The calculation of supports of non vanishing homotopy groups as modules over the fundamental group in terms of the geometry of the locus of non-normal crossings is discussed. We review previous work on the local study of isolated non-normal crossings and relate the motivic zeta function to the local polytopes of quasiadjunction. As an appli…

The numerical equivalence relation for height functions and ampleness and nefness criteria for divisors

In this paper, we study properties of Weil height functions associated with numerically trivial divisors. It helps us to define the fractional limit of $h_E$ with respect to $h_D$ on $U$, with $D$ ample: \[ \Flim_D(E,U) := \liminf_{\substack{P \in U h_D(P) \rightarrow \infty}}\dfrac{h_E(P)}{h_D(P)}. \] The value of $\Flim_D(E,U)$ contains numerical information about a divisor $E$, enough to determine whether $E$ is ample, numerically effective or pseudo-effective.

A local version of the Kawamata-Viehweg vanishing theorem

We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.

Remarks on the non-vanishing conjecture

We discuss a difference between the rational and the real non-vanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem for rationally connected varieties under assuming Shokurov's ACC conjectures.

On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

We investigate the vanishing of $H^1(X,\mathcal{O}_X(-D))$ for a big and nef $\mathbb{Q}$-Cartier $\mathbb{Z}$-divisor $D$ on a log del Pezzo surface $(X,Δ)$ over an algebraically closed field of positive characteristic $p$.

Matrix divisors on Riemann surfaces and Lax operator algebras

Matrix divisors are introduced in the work by A.Weil (1938) which is considered as a starting point of the theory of holomorphic vector bundles on Riemann surfaces. In this theory matrix divisors play the role similar to the role of usual divisors in the theory of line bundles. Moreover, they provide explicit coordinates (Tyurin parameters) in an open subset of the moduli space of stable vector bundles. These coordinates turned out to be helpful in integration of soliton equations. We would like to gain attention to one more relationship between matrix divisors of vector G-bundles (where G is …

Vanishing and identities of conformal blocks divisors

Conformal block divisors in type A on $\bar{M}_{0,{n}}$ are shown to satisfy new symmetries when levels and ranks are interchanged in non-standard ways. A connection with the quantum cohomology of Grassmannians reveals that these divisors vanish above the critical level.

环面簇的 Kawamata–Viehweg 消灭定理

Kawamata-Viehweg vanishing for toric varieties

本文针对射影环面簇上的 klt 配对 $(X,B)$，证明相应的 Kawamata–Viehweg 消灭定理。

Vanishing theorems

We prove some injectivity, torsion-free, and vanishing theorems for simple normal crossing pairs. Our results heavily depend on the theory of mixed Hodge structures on compact support cohomology groups. We also treat several basic properties of semi divisorial log terminal pairs.

Local vanishing and Hodge filtration for rational singularities

Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. We prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization of the structure sheaf along D is at most n-3.

On the non-vanishing conjecture and existence of log minimal models

We prove that the non-vanishing conjecture and the log minimal model conjecture for projective log canonical pairs can be reduced to the non-vanishing conjecture for smooth projective varieties such that the boundary divisor is zero.

Local topology of reducible divisors

We show that the universal abelian cover of the complement to a germ of a reducible divisor on a complex space $Y$ with isolated singularity is $(dimY-2)$-connected provided that the divisor has normal crossings outside of the singularity of $Y$. We apply this result to obtain a vanishing property for the cohomology of local systems of rank one and we also study vanishing in the case of local systems of higher rank. This second version contains a corrected proof of Corollary 4.1 from the first version.

Ext and Tor on two-dimensional cyclic quotient singularities

Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain combinatorial objects using methods from toric geometry. In particular, it is enough to give a combinatorial description of the $\mathop{Ext}\nolimits^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher $\mathop{Ext}\nolimits^i$-groups are then reduced to the case of $\mathop{Ext}\nolimits^1$ via a quive…

Strongly Liftable Schemes and the Kawamata-Viehweg Vanishing in Positive Characteristic

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable, if X and all prime divisors on X can be lifted simultaneously over W_2(k). In this paper, we give some concrete examples and properties of strongly liftable schemes. As an application, we prove that the Kawamata-Viehweg vanishing theorem in positive characteristic holds on any normal projective surface which is birational to a strongly liftable surface.