Karen E. Smith, Howard M. Thompson
Given a singular curve on a smooth surface, we determine which exceptional divisors on the minimal resolution of that curve contribute toward its jumping numbers.
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We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson and Tucker. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.
In this short note we explain how to construct resolutions or regular alterations admitting an ample exceptional divisor, assuming the existence of projective resolutions or regular alterations. In particular, this implies the existence of such resolutions for arithmetic three-dimensional singularities.
We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is…
We prove that, if X is a variety over an uncountable algebraically closed field k of characteristic zero, then any irreducible exceptional divisor E on a resolution of singularities of X which is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs on X centered in Sing X. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs.
We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.
Let X be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of X is a lattice with respect to the Beauville-Bogomolov pairing. A divisor E on X is called a prime exceptional divisor, if E is reduced and irreducible and of negative Beauville-Bogomolov degree. Let E be a prime exceptional divisor on X. We first observe that associated to E is a monodromy involution of the integral cohomology of X, which acts on the second cohomology lattice as the reflection by the cohomology class of E (Theorem 1.1). We then specialize to the case that X is deformation equi…
Let p be a singular point of a variety. Consider a resolution where the preimage of p is a simple normal crossing divisor E. The combinatorial structure of E is described by a cell complex D(E), called the dual graph or dual complex of E. It is known that the homotopy type of D(E) depends only on p, not on the resolution chosen. We prove that this homotopy type can be arbitrary. We also describe which homotopy types can be obtained from rational singularities.
This paper is a sequel to \cite{C}, in which the author studies secant planes to linear series on a curve that is general in moduli. In that paper, the author proves that a general curve has no linear series with exceptional secant planes, in a very precise sense. Consequently, it makes sense to study effective divisors on $\ov{\mc{M}}_g$ associated to curves equipped with secant-exceptional linear series. Here we describe a strategy for computing the classes of those divisors. We pay special attention to the extremal case of $(2d-1)$-dimensional series with $d$-secant $(d-2)$-planes, which ap…
We study the divisorial Zariski decomposition on varieties whose first Chern class is zero. We first prove that any exceptional divisor is contractible (up to a birational map that is an isomorphism in codimension one). We then characterize prime exceptional divisors on holomorphic symplectic manifolds.
We treat threefolds with divisorial contractions whose exceptional divisors contract to compound Du Val points. We prove that general elements in their anticanonical systems around the exceptional divisors have at worst Du Val singularities. As applications to classification, we describe divisorial contractions to compound A_n points, and moreover we conclude that discrepancies of divisorial contractions to compound D_n or E_n points are at most four.
The article is about a "desingularization principle" common to various canonical desingularization algorithms in characteristic zero, and the roles played by the exceptional divisors in the underlying local construction. We compare algorithms of the authors and of Villamayor and his collaborators, distinguishing between the fundamental effect of the way the exceptional divisors are used, and different theorems obtained because of flexibility allowed in the choice of "input data". We show how the meaning of "invariance" of the desingularization invariant, and the efficiency of the algorithm dep…
We find a criterion for an effective divisor $D$ on a smooth surface to be left-orthogonal or strongly left-orthogonal (i.e. for the pair of line bundles $(\mathcal O,\mathcal O(D))$ to be exceptional or strong exceptional).
We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a cA_1 point. We prove that any such contraction is obtained by a suitable weighted blow-up.
Exceptional loci of F-blowups and $G$-Hilbert schemes
研究F吹胀正常分形变的例外极子，通过组合数据给出素除子中心维数公式，并探讨F吹胀与三维终端奇点和canonical奇点的关联。
We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a smooth point. We prove that any such contraction can be obtained by a suitable weighted blow-up.
The moduli stack M_X(E_8) of principal E_8-bundles over a smooth projective curve X carries a natural divisor Delta. We study the pull-back of the divisor Delta to the moduli stack M_X(P), where P is a semi-simple and simply connected group such that its Lie algebra Lie(P) is a maximal conformal subalgebra of Lie(E_8). We show that the divisor Delta induces "Strange Duality"-type isomorphisms between the Verlinde spaces at level one of the following pairs of groups (SL(5), SL(5)), (Spin(8), Spin(8)), (SL(3), E_6) and (SL(2), E_7).
We investigate the case of the Kahler-Ricci flow blowing down disjoint exceptional divisors with normal bundle O(-k) to orbifold points. We prove smooth convergence outside the exceptional divisors and global Gromov-Hausdorff convergence. In addition, we establish the result that the Gromov-Hausdorff limit coincides with the metric completion of the limiting metric under the flow. This improves and extends the previous work of the authors. We apply this to P^1-bundles which are higher-dimensional analogues of the Hirzebruch surfaces. In addition, we consider the case of a minimal surface of ge…
We give a criterion under which a solution g(t) of the Kahler-Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. As t tends to the singular time T from each direction, we prove convergence of g(t) in the sense of Gromov-Hausdorff and smooth convergence away from the exceptional divisors. We call this behavior for the Kahler-Ricci flow a canonical surgical contraction. In particular, our results show that the Kahler-Ricci flow on a projective algebraic surface will perform a sequence of canonical surgical contractions until, in finit…
Let X be the blow-up of a smooth projective 4-fold Y along a smooth curve C and let E be the exceptional divisor. Assume that X is a Fano manifold and has an elementary extremal contraction $ϕ: X \to Z$ of (3,1)-type such that E is $ϕ$-ample (recall that a contraction map for a 4-fold is called (3,1)-type if the exceptional locus is a divisor and its image is a curve). We show that if the exceptional divisor of $ϕ$ is smooth, then Y is isomorphic to $\mathbb{P}^{4}$ and C is an elliptic curve of degree 4.
Hyperbolic Embeddings in Toric Geometry: Effectivity and Deformation Stability
本文研究了托里几何中双曲嵌入的变形行为，通过有效方法分析例外集并证明其变形稳定性。
For a complex analytic function $f$, the exceptional divisor of the jacobian blow-up is of great importance. In this paper, we show what a lemma from the thesis of Lazarsfeld tells one about the structure of this exceptional divisor.
Nash proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that occurs on every resolution. He asked if the converse also holds: does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.
We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric …
We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in terms of the augmented base locus generalizing a result of Keel. We also study nef divisors with positive top intersection number, and discuss some problems related to augmented base loci of log divisors.
Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $α$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of $α$. The negative part of $α$ is then defined as the real effective divisor $N(α)$ whose multiplicity along a prime divisor $D$ is just the generic multiplicity of $α$ along $D$, and we get in that way a divisorial Zariski decomposition of $α$ into the sum of a class $Z(α)$ which is nef in codimension 1 and the class of its negative part $N(α)$, which is exceptio…
In this short note, we extend the results of [Alexeev-Orlov, 2012] about Picard groups of Burniat surfaces with $K^2=6$ to the cases of $2\le K^2\le 5$. We also compute the semigroup of effective divisors on Burniat surfaces with $K^2=6$. Finally, we construct an exceptional collection on a nonnormal semistable degeneration of a 1-parameter family of Burniat surfaces with $K^2=6$.
Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute the classes of such divisors, and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve. By letting the moduli of the curve vary, we study more generally the classes of the related divisors inside the moduli space of stable pointed curves.
We study effective divisors $D$ on surfaces with $H^0(\mathcal O_D)=k$ and $H^1(\mathcal O_D)=H^0(\mathcal O_D(D))=0$. We give a numerical criterion for such divisors, following a general investigation of negativity, rigidity and connectivity properties. Examples include exceptional loci of rational singularities, and spherelike divisors.
We give a myriad of examples of extremal divisors, rigid curves, and birational morphisms with unexpected properties for the Grothendieck--Knudsen moduli space $\bar M_{0,n}$ of stable rational curves. The basic tool is an isomorphism between $M_{0,n}$ and the Brill--Noether locus of a very special reducible curve corresponding to a hypergraph.
We discuss the problem of classifying birational extremal contractions of smooth threefolds where the canonical bundle is trivial along the curves contracted, in the case when a divisor is contracted to a point. We prove the analytic rigidity of the contraction in the case this divisor is normal of degree >= 5 (i.e. we show that the analytic structure of the contraction is completely determined by the isomorphism class of the exceptional locus and its normal bundle in X). This was previously known only in the case of smooth exceptional locus.
We describe the closed cone of moving curves of smooth Fano three- and fourfolds by giving finitely many equations that cut out the cone. The equations are induced by the exceptional divisors of divisorial contractions and by nef divisors on birational models obtained by sequences of flips.The results presented here are part of the author's Ph.D. thesis, written under the supervision of Stefan Kebekus.
This paper deals with the Nash problem, which claims that there are as many families of arcs on a singular germ of surface $U$ as there are essential components of the exceptional divisor in the desingularisation of this singularity. Let $\mathcal{H}=\bigcup \bar{N_α}$ be a particular decomposition of the set of arcs on $U$, described later on. We give two new conditions to insure that $\bar{N_α}\not \subset \bar{N_β}$, $α\not = β$; more precisely,for the first one, we claim that if there exists $f \in {\mathcal{O}}_{U}$ such that $ord_{E_α}(f)<ord_{E_β}(f)$, where $E_α, E_β$ are exceptional d…
Dicritical divisors and hypercurvettes
本文将Abhyankar等人在二维情形下关于双临界分量和曲线的研究推广到任意维数，证明了在任意维光滑簇上，对于给定的双有理映射，存在有理函数使得指定的例外分量具有给定次数的双临界性。
In geometric terms, given a singular foliation of the plane, a dicritical divisor is (whenever it exists) an irreducible component of the exceptional divisor which is transverse to the foliation. Abhyankar gave recently a definition of the dicritical divisors which generalize and algebraicize the geometrical definition in the local case and the polynomial case. Following his work, we give a geometrical interpretation of these dicritical divisors and new proofs of their existence.
The classical McKay correspondence for finite subgroups $G$ of $\SL(2,\C)$ gives a bijection between isomorphism classes of nontrivial irreducible representations of $G$ and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity $\A^2_\C/G$. Over non algebraically closed fields $K$ there may exist representations irreducible over $K$ which split over $\bar{K}$. The same is true for irreducible components of the exceptional divisor. In this paper we show that these two phenomena are related and that there is a bijection between nontrivial irreduc…
We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of th…
Automorphisms of Hilbert Schemes of Points on Abelian Surfaces
本文构造 Abel 曲面任意点数 Hilbert 概形上保持例外除子的非自然自同构，并证明 Picard 秩 1 的主极化或极化自交为平方数时两点情形的所有自同构均自然。
We give a conjectural description for the cone of effective divisors of the Grothendieck-Knudsen moduli space of stable rational curves with n marked points. Namely, we introduce new combinatorial structures called hypertrees and show they give exceptional divisors with many remarkable properties.
To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated …
We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface $X$ in certain families of surfaces of general type with $p_g=0$ and $K_X^2=3,4,5,6,8$. We also compute the algebra of derived endomorphisms for an appropriately chosen exceptional collection, and the Hochschild cohomology of the corresponding quasiphantom category. As a consequence, we see that the subcategory generated by the exceptional collection does not vary in the family of surfaces. Finally, we describe the semigroup of effective divisors on each …
In a previous work we have introduced and studied the notion of embedded $\mathbf{Q}$-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we explicitly compute an embedded $\mathbf{Q}$-resolution of a Yomdin-Lê surface singularity $(V,0)$ in terms of a (global) embedded $\mathbf{Q}$-resolution of their tangent cone by means of just weighted blow-ups at points. The generalized A'Campo's formula in this setting is applied so as to compute the characteristic polynomial. As a consequence, an exceptional divisor in the resolutio…
Blow-Up Constructions and Applications to Segre Classes and Multidegree Formulas
通过比较同时多重分次爆破与迭代爆破，建立例外除子的双有理对应，并应用于Segre类的乘积公式和多射影空间闭子簇的度公式。
Let $X$ be a fixed projective scheme which is flat over a base scheme $S$. The association taking a quasi-projective $S$-scheme $Y$ to the scheme parametrizing $S$-morphisms from $X$ to $Y$ is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide …
We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M_{10} consisting of curves sitting on K3 surfaces. This divisor is not of classical Brill-Noether type, and is known to give a counterexample to the Slope Conjecture. The computation relies on the fact that this divisor has four different incarnations as a geometric subvariety of the moduli space of curves, one of them as a higher rank Brill-Noether divisor consisting of curves with an exceptional rank 2 vector bundle. As an application we describe the birational nature of the mod…
We classify extremal divisorial contraction which contracts a divisor to a curve from a smooth fourfold. We prove the exceptional divisor is $\Bbb P^2$bundle or quadric bundle over a smooth curve and the contraction is the blowing up along the curve.
We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which…
We describe a systematic way of constructing effective divisors on the moduli space of stable curves of genus g having exceptionally small slope. We prove that any divisor on \bar{M}_g consisting of curves failing a certain Green-Lazarsfeld syzygy type condition, provides a counterexample to the Harris-Morrison Slope Conjecture. These divisors generalize our original isolated counterexample to the Slope Conjecture which was the divisor on M_{10} of curves lying on K3 surfaces. We also introduce a new stratification of M_g, somewhat similar to the classical stratification given by gonality, but…
Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $π: X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $π$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.
The paper consists of three parts. In the first of them different kinds stability are discussed. In particular, the stability concept with respect to nef divisor is introduced. A structure of rigid and superrigid vector bundles on smooth projective surfaces with nef anticanonical class is studied in the second part. We prove that any superrigid bundle has a unique exceptional filtration. In the last part we give constructive description of exceptional bundles on these surfaces.
Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of ``M-theory'') and a four-dimensional physical theory (using the ``F-theory'' construction). A key issue in both theories is the calculation of the ``superpotential''of the theory. We propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contributes. We then apply our techniques in explicit examples, in particular, when the base B of the elliptic fibration is a toric variety or a Fano 3-fold. When B is Fano, we…
We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated to a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete 2-level structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.
For $m\in \IN, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a toric surface $S.$ For $m$ big enough, we connect the number of a class of these irreducible components to the number of exceptional divisors on the minimal resolution of $S.$
In this paper, we study pathologies of Du Val del Pezzo surfaces defined over an algebraically closed field of positive characteristic by relating them to their non-liftability to the ring of Witt vectors. More precisely, we investigate the condition (NB): all the anti-canonical divisors are singular, (ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type, Picard rank, and anti-canonical degree, (NK): there exists an ample $\mathbb{Z}$-divisor which violates the Kodaira vanishing theorem for $\mathbb{Z}$-divisors, and (NL): the pair $(Y, E)$ do…
The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but their geometric interpretation is still mysterious. We show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is an exceptional group of type E_6. This is the first known exceptional case, and it suggests a connection with the monodromy of the Gauss map.
Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot π^*L_S - α\cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $π:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide a criterion for $k$-very ampleness of $M$. Also, we deal with the case when $S$ is an arbitrary surface of Picard number one with a numerically trivial canonical divisor.
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…
Given a smooth complex projective surface $S$ and an ample divisor $H$ on $S$, consider the blow up of $S$ along $k$ points in general position. Let $H'$ be the pullback of $H$ and $E_1,..., E_k$ be the exceptional divisors. We show that $L = nH' -E_1 - ... -E_k$ is ample if and only if $L^2$ is positive provided the integer $n$ is at least 3.
This is a paper based on a talk given at the Warwick Symposium on Algebraic Geometry in 1996. The resolution of singularities given by F. Bogomolov and A. Pantev (arXiv:math.AG/9603019) is presented in a self-contained and "elementary" manner. The main result is that given a variety and a proper closed subvariety, there is a projective birational morphism from a smooth variety to the given variety so that the inverse image of the given closed subvariety and the exceptional locus are sub-divisors of a divisor with simple normal crossings.
Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces
本文研究Hirzebruch曲面上由无穷远处非正除子赋值定义的旗，计算相应Seshadri型常数并显式确定Newton-Okounkov体的顶点，证明其为四边形或三角形。
Functorial resolution by torus actions
提出利用环面作用在带SNC除子的光滑簇上，通过新引入的沿光滑加权中心的配边爆破，实现特征零下簇的快速嵌入解消与理想主化，并应用于正和混合特征奇点的解消。
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