Singularities of dual varieties in characteristic 2

Singularities of Affine Schubert Varieties

This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\mathrm{A}^{(1)}_\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.

Characterizing Projective Spaces for Varieties with at Most Quotient Singularities

We generalize the well-known numerical criterion for projective spaces by Cho, Miyaoka and Shepherd-Barron to varieties with at worst quotient singularities. Let $X$ be a normal projective variety of dimension $n \geq 3$ with at most quotient singularities. Our result asserts that if $C \cdot (-K_X) \geq n+1$ for every curve $C \subset X$, then $X \cong \PP^n$.

Singularities of Restriction Varieties in $OG(k, n)$

Restriction varieties in the orthogonal Grassmannian are subvarieties of $OG(k, n)$ defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular, Schubert varieties of Type B and D are examples of restriction varieties. In this paper, we introduce a resolution of singularities for restriction varieties in $OG(k, n)$, and give a description of their singular locus by studying components of the exceptional locus of the resolution.

Singularities of generalized Richardson varieties

Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this paper, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singula…

Singularities of divisors of low degree on simple abelian varieties

It is known by results of Kollár, Ein, Lazarsfeld, Hacon and Debarre that divisors representing principal and other low degree polarizations on abelian varieties have mild singularities. In this note we extend such results to polarizations of degree $<g$ on simple $g$-dimensional abelian varieties, settling a conjecture of Debarre and Hacon.

Singularities of secant varieties

We study the singularities of the secant variety $Σ(X,L)$ associated to a smooth variety $X$ embedded by a sufficiently positive adjoint bundle $L$. We show that $Σ(X,L)$ is always Du Bois singular. Examples of secant varieties with worse singularities when $L$ has weak positivity are provided. We also give a necessary and sufficient condition for $Σ(X, L)$ to have rational singularities.

Resolution of singularities of complex algebraic varieties and their families

We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties, leading to joint work on toroidal orbifolds with Michael Temkin and Jarosław Włodarczyk.

Generic singularities of Schubert varieties

We describe the generic singularity of a Schubert variety of type A on each irreducible component of its singular locus. This singularity is given either by a cone of rank one matrices, or a quadratic cone.

Singularities of divisors of low degree on abelian varieties

Building on previous work of Kollar, Ein, Lazarsfeld, and Hacon, we show that ample divisors of low degree on an abelian variety have mild singularities in case the abelian variety is simple or the degree of the polarization is two.

The global indices of log Calabi-Yau varieties --A supplement to Fujino's paper: The indices of log canonical singularities--

This paper gives the all possible global indices of log Calabi-Yau 3-folds with standard coefficients on the boundaries and having lc, non-klt singularities. This follows easily from the discussion in the paper: The indices of log canonical singularities by Fujino (AG/9909035).

A Griffiths' Theorem for varieties with isolated singularities

By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.

The Moduli Space of Abelian Varieties and the Singularities of the Theta Divisor

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

阿贝尔簇上小度数除子的奇点

Singularities of divisors of small degrees on abelian varieties

本文将 Pareschi 关于单阿贝尔簇上小度数丰富除子具有温和奇点的结果推广到一般极化阿贝尔簇，并进一步证明度数为 3 或 4 的丰富除子也具有温和奇点。

关于具有轻微奇点的射影正常簇的Eisenbud-Goto猜想

The Eisenbud-Goto conjecture for projectively normal varieties with mild singularities

本文证明了对于具有特定奇点类型的2-非常 ample射影正常簇，Eisenbud-Goto猜想成立。

Singularities of log varieties via jet schemes

We study logarithmic jet schemes of a log scheme and generalize a theorem of M. Mustata from the case of ordinary jet schemes to the logarithmic case. If X is a normal local complete intersection log variety, then X has canonical singularities if and only if the log jet schemes of X are irreducible.

从Hodge理论视角看割线簇的奇点

Singularities of secant varieties from a Hodge theoretic perspective

本文从Hodge理论视角，利用双有理几何方法研究光滑射影簇割线簇的奇点，给出了p-Du Bois奇点的充要条件，并证明其奇点非更高有理，进而推导出Kodaira-Akizuki-Nakano型消失定理。

Singularities of the biextension metric for families of abelian varieties

In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$.

Singularities of Cox Rings of Fano Varieties

Let $X$ be a smooth complete Fano variety over $\mathbb{C}$. We show that the Cox ring $\bigoplus_{L\in\text{Pic}(X)}H^0(X,\mathcal{O}_X(L))$ is Gorenstein with canonical singularities.

Singularities of Fano varieties of lines on singular cubic fourfolds

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.

Singularities and syzygies of secant varieties of nonsingular projective curves

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfie…

光滑射影簇的割线簇的奇点与合冲

Singularities and syzygies of secant varieties of smooth projective varieties

本文研究光滑射影簇的高阶割线簇，证明曲面情形在正线丛下具有Du Bois奇点及线性合冲，并揭示其奇点类型由曲面结构层上同调决定，同时给出高维低阶情形的类似结果及高阶情形的否定性结论。

Codimension Two Determinantal Varieties with Isolated Singularities

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…

Stringy E-functions of varieties with A-D-E singularities

The stringy E-function for normal irreducible complex varieties with at worst log terminal singularities was introduced by Batyrev. It is defined by data from a log resolution. If the variety is projective and Gorenstein and the stringy E-function is a polynomial, Batyrev also defined the stringy Hodge numbers as a generalization of the Hodge numbers of nonsingular projective varieties, and conjectured that they are nonnegative. We compute explicit formulae for the contribution of an A-D-E singularity to the stringy E-function in arbitrary dimension. With these results we can say when the stri…

Singularities of pluri-fundamental divisors on Gorenstein Fano varieties of coindex $4$

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.

Pushforwards of Measures on Real Varieties under Maps with Rational Singularities

Let $X,Y$ be algebraic varieties defined over $\Bbb R$. Assume $Y$ is smooth and $X$ is Gorenstein. Suppose $φ:X\to Y$ is a flat $\Bbb R$-morphism such that all the fibers have rational singularities. We show that the pushforward of any smooth, compactly supported measure on $X$ has a continuous density with respect to any smooth measure with non-vanishing density on $Y$. This extends a result of Aizenbud and Avni from the $p$-adic case to the archimedean case.

Singularities of divisors on flag varieties via Hwang's product theorem

We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety $X=G/P$ and an effective $\mathbb{Q}$-divisor $D$ stable with respect to a Borel subgroup the pair $(X,D)$ is Kawamata log terminal if and only if $\lfloor D\rfloor=0$.

具有预k-有理奇点的代数簇上的微分形式

Differential forms on varieties with pre-$k$-rational singularities

本文针对具有预k-有理奇点的复代数簇，比较两类紧支上同调群的混合霍奇结构，研究其微分形式相关概念，重点考察Du Bois复形分次部分的深度。

紧化Kleinian奇点上的箭图簇与带标架层

Quiver Varieties and Framed Sheaves on Compactified Kleinian Singularities

该研究针对Kleinian奇点$\mathbb{C}^2/\Gamma$，引入带光滑栈式除子的自然紧化栈，证明其上满足特定条件的带标架层集合与一类Nakajima箭图簇相关，部分推广了此前点Hilbert概型的相关研究。

On the ideals and singularities of secant varieties of Segre varieties

We find generators for the ideals of secant varieties of Segre varieties in several the cases including the Garcia-Stillmann-Ssturmfels conjecture for four factors and prove results about their singularities.

具有孤立有理奇点的射影簇的半正交分解

Semiorthogonal decompositions of projective varieties with isolated rational singularities

本文研究具有孤立有理奇点的射影簇，提出从小消解的半正交分解诱导目标分解的方法，并通过零范畴生成元与消解态射逆像的性质证明其存在性，推广了奇异曲面的已有结果。

Smoothings of Fano varieties with normal crossing singularities

This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of logarithmic structures and deformations is used in order to prove that X is smoothable by a smooth variety, if and only if T^1(X)=O_D, where D is the singular locus of X.

Jet schemes and singularities

We give a self-contained presentation of the basic results on jet schemes of singular varieties. Applications are given to invariants of singularities, such as minimal log discrepancies. We simplify our older approach to Inversion of adjunction for locally complete intersection varieties, and we prove a more general result for arbitrary Q-Gorenstein varieties embeded in nonsingular varieties (this statement has been proved independently also by Kawakita) .

On the Singularities of Degenerate Abelian Varieties

In 2002, Alexeev provided a modular interpretation for the toroidal compactification of $A_g$ for the 2nd Voronoi fan. In this paper we show that pairs $(X,Θ)$ in the boundary of this moduli space are semi-log canonical, the analog of log canonical in the non-normal setting. This extends the result of Koll${á}$r stating that principally polarized abelian pairs $(X,Θ)$ are log canonical.

Modular quotient varieties and singularities by the cyclic group of order $2p$

We classify all $n$-dimensional reduced Cohen-Macaulay modular quotient variety $\mathbb{A}_\mathbb{F}^n/C_{2p}$ and study their singularities, where $p$ is a prime number and $C_{2p}$ denotes the cyclic group of order $2p$. In particular, we present an example that demonstrates that the problem proposed by Yasuda \cite[Problem 6.6]{Yas2015} has a negative answer if the condition that "$G$ is a small subgroup" was dropped.

Cartier and Weil Divisors on Varieties with Quotient Singularities

The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.

Surface singularities dominated by smooth varieties

We give a version in characteristic $p>0$ of Mumford's theorem characterizing a smooth complex germ of surface $(X,x)$ by the triviality of the topological fundamental group of $U=X\setminus \{x\}$. This note relies on discussions the authors had during the Christmas break 2009/10 in Ivry. They have been written down by Hélène in the night when Eckart died, as a despaired sign of love.

Rational singularities of normal T-varieties

A T-variety is an algebraic variety X with an effective regular action of an algebraic torus T. Altmann and Hausen gave a combinatorial description of an affine T-variety X by means of polyhedral divisors. In this paper we compute the higher direct images of the structure sheaf of a desingularization of X in terms of this combinatorial data. As a consequence, we give a criterion as to when a T-variety has rational singularities. We also provide a partial criterion for a T-variety to be Cohen-Macaulay. As an application we characterize in this terms quasihomogeneous elliptic singularities of su…

全局平滑奇异Fano和Calabi-Yau品种

Global smoothing of singular Fano and Calabi-Yau varieties

本文研究了Fano和Calabi-Yau品种的全局平滑问题，证明了在特定奇点条件下，这些品种可以通过变形获得更光滑的结构，并推广了已有结论。

Singularities of non-$\mathbb{Q}$-Gorenstein varieties admitting a polarized endomorphism

In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and Höring.

On weak Fano varieties with log canonical singularities

We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-…

曲线割线簇奇点的不变量

Invariants of the singularities of secant varieties of curves

本文研究光滑射影曲线的 $k$-平面割线簇的奇点不变量，计算了任意曲线的交点上同调，并证明有理正规曲线的所有割线簇均为满足 Poincaré 对偶的有理同调流形，同时计算了最大非平凡割线簇的邻近圈与消没圈。

有限商奇点的 $K$-正则性

$K$-regularity for finite quotient singularities

本文研究有限商奇点的$K$-正则性，证明在奇点轨迹含于仿射闭子簇时$K_1$-正则性等价于光滑性，并构造反例表明一般情形不成立，通过肯定Fogarty问题获得判据。

Singularities of generic projection hypersurfaces

Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -- the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.

Log canonical thresholds on varieties with bounded singularities

We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition.

Differential forms in positive characteristic avoiding resolution of singularities

This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than Kähler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with Kähler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number o…

Relative Bott-Samelson varieties

We prove that, defined with respect to versal flags, the product of two relative Bott-Samelson varieties over the flag bundle is a resolution of singularities of a relative Richardson variety. This result generalizes Brion's resolution of singularities of Richardson varieties to the relative setting. It reflects the phenomenon that the local geometry of a relative Richardson variety is completely governed by the two intersecting relative Schubert varieties, studied by Chan-Pflueger. We also prove an analogous theorem in the case of relative Grassmannian Richardson varieties, thereby furnishing…

Singularities of linear systems and boundedness of Fano varieties

We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…

带孤立奇点的簇的导出分层与算术相交理论

Derived Stratifications and Arithmetic Intersection Theory for Varieties with Isolated Singularities

本文针对带孤立奇点的簇，用导出几何等工具建立分层德拉姆理论，统一拓展相关理论框架，并填补Ohsawa证明的缺口，验证Cheeger-Goresky-MacPherson猜想的相关结论。

Kawamata log terminal singularities of full rank

We study Kawamata log terminal singularities of full rank, i.e., $n$-dimensional klt singularities containing a large finite abelian group of rank $n$ in its regional fundamental group. The main result of this article is that klt singularities of full rank degenerate to cones over log crepant equivalent toric quotient varieties. To establish the main theorem, we reduce the proof to the study of Fano type varieties with large finite automorphisms of full rank. We prove that such Fano type varieties are log crepant equivalent toric. Furthermore, any such Fano variety of dimension $n$ contains an…

Singularities on Demi-Normal Varieties

The birational classification of varieties inevitably leads to the study of singularities. The types of singularities that occur in this context have been studied by Mori, Kollár, Reid, and others, beginning in the 1980s with the introduction of the minimal model program. Normal singularities that are terminal, canonical, log terminal, and log canonical, and their non-normal counterparts, are typically studied using a resolution of singularities (or a semiresolution) and finding numerical conditions that relate the canonical class of the variety to that of its resolution. In order to do this, …

Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations…

簇的分次格罗滕迪克环与$\boldsymbol{\u0394}$-奇点上的对合

Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities

该研究将簇的分次格罗滕迪克环实现为特定子环的二次扩张，引入对合$\u0394$并证明其与对称幂运算的可交换性，还研究$\u0394$-奇异簇并应用于簇紧化与Kapranov ζ函数无理性问题。

阿贝尔簇上基点的奇点

Singularities of base loci on abelian varieties

本文证明复阿贝尔簇上完全线性系统的基理想的log规范阈值≥1，且等式成立当且仅当基点有除子分量。

On Semirational Singularities

We study semiresolutions of quasi-projective varieties with properties $G_1$, $S_2$, and seminormality. Equivalently, these are varieties $X$ with Serre's $S_2$ property, such that there exists an open subvariety $U$, with complement of codimension at least two, such that the only singularities of $U$ are (analytically) double normal crossings. Such varieties have been called "demi-normal" by Kollár \cite{Kol13}. First, we discuss why these properties are ideal for the study of nonnormal varieties that appear in the birational classification of varieties. We define semiresolutions and provide …