Number of singular points on projective surfaces

具有孤立奇点的射影超曲面的Milnor数

Milnor numbers of projective hypersurfaces with isolated singularities

设V是固定次数和维数的射影超曲面，且只有孤立奇点。我们证明，如果在V的奇点的Milnor数的总和是大的，那么V不能有一个点的大的重数，除非V是一个锥。作为应用，我们对Dimca和Papadima的一个猜想给出了肯定的回答.

Moduli of regular singular parabolic connections of spectral type on smooth projective curves

We define a moduli space of stable regular singular parabolic connections of spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlevé property of the isomonodromic deformation.

加权射影空间中光滑完全交的量子上同调与奇异环面簇

Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties

我们推广Givental定理完全相交光滑环面品种在Fano的情况。特别是，我们发现Gromov-Witten不变量的Fano品种的尺寸$\geq 3$，这是完全的交叉加权射影空间和奇异环面品种。我们将Riemann-Roch方程推广到加权射影空间。我们计算计数矩阵的光滑Fano三倍皮卡德群Z和anticanonical度2，8，和16。

Variety of Singular Quadrics Containing a Projective Curve

We study the variety of rank $\leq k$ quadrics containing a general projective curve and show that it has the expected dimension in the range $g-d+r\leq 1$. By considering the loci where this expectation is not true, we construct new divisor classes in $\overline{\mathcal{M}}_{g,n}$. We use one of these classes to show that $\overline{\mathcal{M}}_{15,9}$ is of general type.

射影超曲面的不变量与刚性

Invariants and rigidity of projective hypersurfaces

本文继续我们的研究\cite{DS1, DS2, DS3}，计算了与Milnor代数相关的基于Hilbert-Poincaré级数的不变量.我们的计算是针对一些经典的曲面和3重不同配置的孤立奇点。作为E. Sernesi，我们给出了经典超曲面的例子是（或不是）射影刚性。我们还包括一个奇异的程序来计算不变量，并决定如果一个奇异的射影超曲面是节点和射影刚性。

立方超曲面的奇点与射影化切丛的超平面截面

Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space

研究立方超曲面的奇点性质，并通过线性代数方法描述射影空间射影化切丛的超平面截面，扩展了Mazouni-Nagaraj的高维结果。

具有唯一奇点和最大代数重数的射影叶状结构

Projective Foliations with a Unique Singular Point and Maximum Algebraic Multiplicity

本文研究复射影平面上具有唯一奇点的余维一全纯叶状结构，聚焦次数为$d$且奇点最大重数为$d$的叶状结构，明确描述该类结构、刻画具唯一奇点的叶状结构，并给出其希尔伯特-芒福德稳定性的相关结果。

射影超曲面成为 GIT（半）稳定的若干充分条件

Several sufficient conditions for projective hypersurfaces to be GIT (semi)stable

本文研究射影超曲面的 GIT 稳定性，依据维数、次数、奇异轨迹及奇点重数给出充分条件，并在孤立二重奇点情形下利用 Hessian 矩阵秩进行判定。

具有终端奇点的projective varieties上的长Fano-Mori压缩

Fano-Mori contractions of high length on projective varieties with terminal singularities

设X是一个具有终端奇点的射影簇，L是X上的一个充分Cartier因子.证明了：如果f是一个与极值射线$ R \subset \bar {NE(X)}$相关联的双有理压缩，使得R.（K_X+（n-2）L）<0，则f是光滑点的加权blow-up。然后，我们分类与极值射线R相关联的除数收缩，使得R。（K_X+rL）<0，其中r是非负有理数，并且f的纤维具有小于或等于r+1的维数。

${\mathbb P}^n$的平面变形

Flat deformations of ${\mathbb P}^n$

本文研究了射影空间的射影平坦变形。证明了射影空间的射影平坦变形的奇异纤维要么出现在余维1上，要么出现在基的奇点上。我们还描述了具有光滑全空间的射影空间的射影平坦变形，并讨论了平坦性准则。

射影平面的叶理不变性理论

Invariant theory of foliations of the projective plane

研究了射影平面上奇异叶理的不变量理论。我们的第一个主要结果是：一个m>1次的叶理是不稳定的，仅当它在1维上有奇点或包含一个重数至少为（m^2-1）/（2 m +1）的孤立奇点.我们的第二个主要结果是构造了一个从m次叶理空间到m^2+m-2次曲线空间的不变映射。我们在m=2的情况下明确地描述这个映射。

奇异簇的圈映射的满射性

Surjectivity of cycle maps for singular varieties

Jannsen的一个定理断言，如果一个光滑投射簇有内射圈类映射，它就有满射圈类映射。本文的目的是给出奇异拟射影簇的Jannsen定理的一个版本。

具有任意示性数集的奇异projective variety的存在性

Existence of a singular projective variety with an arbitrary set of characteristic numbers

已知紧致复流形的陈示性数不可能具有任意值。它们满足一定的可除性条件。W. Ebeling和S. M. Gusein-Zade给出了奇异紧复解析簇的Chern示性数的定义。证明了存在一个具有任意示性数集的奇异射影簇。

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof…

Classical formulae on projective surfaces and $3$-folds with ordinary singularities, revisited

As an application of universal polynomials for local and multi-singularities of maps, we revisit classical enumerative formulae of Salmon-Cayley-Zeuthen for projective surfaces and analogous formulae of Segre-(B.)Severi-Roth for projective $3$-folds. In particular, several examples of actual computation are given using universal polynomials for computing weighted Euler characteristics of singularity loci.

Int-amplified endomorphisms on normal projective surfaces

We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.

Rational homotopy of complex projective varieties with normal isolated singularities

Let X be a complex projective variety of dimension n with only isolated normal singularities. In this paper we prove, using mixed Hodge theory, that if the link of each singular point of X is (n-2)-connected, then X is a formal topological space. This result applies to a large class of examples, such as normal surface singularities, varieties with ordinary multiple points, hypersurfaces with isolated singularities and more generally, complete intersections with isolated singularities. We obtain analogous results for contractions of subvarieties.

Vanishing homology of projective hypersurfaces with 1-dimensional singularities

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.

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。

Moishezon空间的射影性判据与射影辛簇的稠密性

Projectivity criterion of Moishezon spaces and density of projective symplectic varieties

一个奇异的Moishezon空间即使是Kaehler空间也不是投射的。本文给出了Kaehler-Moishezon空间的一个射影判据。我们将使用这个射影准则，以显示密度的投影辛品种在Kuranishi家庭的（奇异）辛品种。

任意特征上具有典范奇点的曲面的Bogomolov-Gieseker-Koseki不等式

The Bogomolov-Gieseker-Koseki inequality on surfaces with canonical singularities in arbitrary characteristic

本文将Bogomolov-Gieseker不等式从特征零光滑射影曲面推广到任意特征、具典范奇点的射影曲面，并推广其若干经典应用。

关于非射影法曲面

On non-projective normal surfaces

我们构造了非投射正规真代数曲面的例子，并讨论了它们的Neron-Severi群的病态行为。我们的曲面是关于一条射影直线和一条高亏格曲线的乘积的双有理曲面。

（1，2d）型交换曲面的射影正规性

Projective normality of abelian surfaces of type (1,2d)

我们证明了由非常丰富的（1，2 d）型线丛L嵌入P^N的阿贝尔曲面是射影正规的当且仅当d>=4。这就完成了对嵌入完全线性系的交换曲面的射影正规性的研究。

数值平凡纤维化

Numerical trivial fibrations

本文对光滑射影簇上具有正曲率流的奇异hemitian线丛及其上的不可约曲线建立了一个相交理论。证明了奇异厄米线丛的自然有理纤维化结构的存在性。对于光滑射影簇上的任意伪有效线丛，我们证明了存在与线丛相联系的有理自然纤维化结构。我们还刻画了光滑射影簇上数值平凡的奇异Hermitain线丛。

Projective Automorphism Groups of Nonsingular Quartic Surfaces

For every p >= 5, we determine all Z_p-invariant nonsingular quartic surfaces in the three dimensional projective space over an algebraically closed field of characteristic zero. In some cases, we also determine their full projective automorphism groups.

Automorphisms of quasi-projective surfaces over fields of finite characteristic

We prove that the group of automorphisms of any quasi-projective surface $S$ in finite characteristic has the $p$-Jordan property.

有理曲面的奇异性计算

Computing the Singularities of Rational Surfaces

给出了一个有理射影曲面$\cS$的有理射影参数化$\cP(\ttt,\sss,\vvv)$，我们提出了一个算法，使得除了$\cP$的射影基点的有限集（可能为空）$\cB$外，将射影参数平面分解为$\projdos\setminus \cB=\cup_{k=1}^{\ell} \cSm_k$，使得如果$(\ttt_0:\sss_0:\vvv_0)\in \cSm_k$，则$\cP(\ttt_0,\sss_0,\vvv_0)$是重数$k$的$\cS$的一个点.

完备相交曲面上的射影曲线的亏格

The genus of projective curves on complete intersection surfaces

本文用射影曲线的次数和曲面上定义方程的次数来界定完备交曲面上的射影曲线的亏格。

Chirality of Real Non-singular Cubic Fourfolds and Their Pure Deformation Classification

In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.

具有结点奇点的奇异射影曲线上的相干层

Coherent Sheaves on Singular Projective Curves with Nodal Singularities

我们给出了这个问题的完整答案：在哪些曲线上相干层范畴$\Coh_{X}$是驯服的。答案是：这些只是Drozd-Greuel列表中的曲线。此外，在这种情况下，导出范畴$D^{-}(\Coh_{X})$也是驯服的。我们给出这个范畴的对象以及范畴$D^{b}(\Coh_{X})$、$\Coh_{X}$的对象的明确描述。在凝聚层中，我们描述了向量丛、无挠层、混合层和摩天大楼层。

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。

On the formal principle for curves on projective surfaces

We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.

Smooth complex projective rational surfaces with infinitely many real forms

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.

具有整数Zariski分解的射影K3曲面

Integral Zariski decompositions on smooth projective surfaces II

本文通过一个显式的有限步终止算法，刻画了所有使得每个整数伪有效除子都具有整数Zariski分解的射影K3曲面。

关于射影四维空间中有理曲面的注记

A note on rational surfaces in projective four-space

证明了在射影四维空间中，以三次或四次为界的光滑有理曲面的次数不超过12。本文还证明了在射影四维空间中，由具有“简单基点“（见正文）的线性系象Fn的光滑有理曲面的次数至多为12。

关于奇异EPW立方体的几何

On the geometry of singular EPW cubes

本文研究奇异EPW立方体的几何，借鉴O'Grady处理双EPW六次超曲面的方法，成功构造了奇异EPW立方体的射影光滑小消解。

Zeta function of projective hypersurfaces with ADE singularities

Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…

射影曲面上的网和线的外因子

Extactic divisors for webs and lines on projective surfaces

给定一个网（多叶）和一个线性系在一个射影曲面上，我们构造因子切除轨迹，其中某些元素的线性系有异常接触的网页的叶子。我们应用这些想法重新获得一个经典的结果鲑鱼的数量上的线的射影曲面。在不同的静脉，我们调查的数量的线和不相交的线包含在一个射影曲面和切线的接触分布。

光滑射影曲面上的整Zariski分解

Integral Zariski decompositions on smooth projective surfaces

该研究针对代数几何中的除子分解问题，刻画了满足所有整伪有效除子都存在整Zariski分解的光滑射影曲面的特征。

Characteristic classes and stability conditions for projective Kleinian orbisurfaces

We construct Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the Toën-Hirzebruch-Riemann-Roch theorem.

射影四维空间中12次有理曲面的构造（附Kristian Ranestad）

Construction of rational surfaces of degree 12 in projective fourspace (with an appendix by Kristian Ranestad)

本文的目的是给出射影四维空间中两种不同的具有12次和13个截亏格的光滑有理曲面的构造。特别是，我们建立了五个不同的家庭的存在性光滑的理性曲面在射影四维空间与规定的不变量。

多结点射影曲面

Projective surfaces with many nodes

本文推广了I.多尔加切夫Mendes Lopes和R.帕迪尼证明了光滑的射影复曲面$X$（不一定是极小的）包含$h^{1,1}(X)-1$不相交的$(-2)$ -曲线当且仅当$X$同构于极小有理直纹曲面${\bf F}_2$或${\bf P}^2$或伪射影平面.我们还描述了光滑的射影复曲面$X$与$h^{1,1}(X)-2$不相交的$(-2)$ -曲线。

射影超曲面强欧拉齐性的合冲秩刻画

A Syzygy Rank Characterization of Strongly Euler Homogeneity for Projective Hypersurfaces

该研究利用雅可比合冲给出射影超曲面强欧拉齐性奇点的合冲秩刻画，改进了孤立奇点情形下的已有结果，并将其推广到光滑射影环面簇。

A note on a smooth projective surface with Picard number 2

We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj-Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103--108].

射影曲面的稳定性条件与双有理几何

Stability conditions and birational geometry of projective surfaces

我们表明，极小模型程序在任何光滑的射影表面上实现的Bridgeland稳定对象的模空间的变化的导出范畴的相干层。

The automorphism group of a rational projective k*-surface

We consider possibly singular rational projective k*-surfaces and provide an explicit description of the unit component of the automorphism group in terms of isotropy group orders and intersection numbers of suitable invariant curves. As an application, we characterize the almost homogeneous rational projective k*-surfaces and we specify the two-dimensional groups acting almost transitively as well as the corresponding homogeneous spaces.

关于光滑复射影超曲面的对偶

On the duals of smooth projective complex hypersurfaces

本文证明一般光滑复射影超曲面的对偶至少含有一个重数不低于维数的非常奇点，并给出关键构造。

关于射影空间的特殊二次双有理变换

On special quadratic birational transformations of a projective space

一个从射影空间到一个不太奇异的射影簇的双有理映射具有一个不可约的非奇异基轨迹方案（特殊的双有理变换）是一种罕见的现象，足以使分类结果有意义和简洁。我们将集中讨论由二次方程定义的变换到某些簇（特别是小度数的射影超曲面）上，令人惊讶的是，基轨迹是出现在其他上下文中的有趣的射影流形;例如，附加理论的例外，小度数或小余维流形，Severi或更一般的齐次簇。特别是，我们将分类：到二次超曲面的二次二次变换;到del Pezzo簇的四次三次变换;基轨迹（方案）的维度最多为三的变换。

A singular Darboux type theorem and non-integrable projective distributions of degree one

We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on projective spaces.

Bounding cohomology on a smooth projective surface

The following conjecture arose out of discussions between B. Harbourne, J. Roé, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal O_X(C))$ for every prime divisor $C$ on $X$. We show that the conjecture is true for some smooth projective surfaces with Picard number 2.

有限域上射影$3$ -空间中曲面的点数

Numbers of points of surfaces in the projective $3$-space over finite fields

在前一篇文章中，我们建立了${\Bbb F}_q$上射影$3$ -空间中曲面点数的初等界。在本文中，我们给出了达到初等界的曲面的完整列表。确切地说，这些曲面是双曲曲面、非奇异厄米曲面和包含$3$ -空间的所有${\Bbb F}_q$ -点的最小度曲面。

Counting lines on projective surfaces

We prove a bound on the number of lines on a smooth degree-d surface in three-dimensional projective space for $d \geq 3$. This bound improves a bound due to Segre and renders some of his arguments rigorous. It is the best known bound for $d \geq 6$.

Logarithmic forms and singular projective foliations

In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$ with some extra degree assumptions. We determine new irreducible components of the moduli space of codimension two singular projective foliations of any degree, and we show that they are generically reduced in their natural scheme structure. Our method is based on an explicit description of the Zariski tangent space of the corresponding moduli space at a given…

加权射影三维空间中log del Pezzo曲面上的Kahler-Einstein度量

Kahler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces

我们确定了在加权射影3-空间中反正则嵌入的拟光滑log del Pezzo曲面的完整列表。我们证明，其中许多承认Kähler-Einstein度量，其中大多数没有老虎。