Local cohomologies of isolated non F-rational singularities

Invariants of Artinian Gorenstein Algebras and Isolated Hypersurface Singularities

We survey our recently proposed method for constructing biholomorphic invariants of quasihomogeneous isolated hypersurface singularities and, more generally, invariants of graded Artinian Gorenstein algebras. The method utilizes certain polynomials associated to such algebras, called nil-polynomials, and we compare them with two other classes of polynomials that have also been used to produce invariants.

$F$-幂零性与孤立奇点之外的Hodge滤过

$F$-nilpotence and Hodge filtrations beyond isolated singularities

本文通过Hodge理论刻画特征零簇的开$F$-幂零型与开弱$F$-幂零型，将Srinivas-Takagi结果推广至非孤立奇点情形，并分别无条件及在弱普通性猜想下证明两个方向。

检测和量化离散赋值环上的孤立奇点

Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings

本文在混合特征(0,p)下发展孤立超曲面奇点理论，研究离散赋值环上的商环，引入经典Tjurina和Milnor数的类比，证明确定性定理和Mather-Yau定理的推广，并定义数值不变量以区分非可分和可分情况下的孤立奇点。

Face numbers of pseudomanifolds with isolated singularities

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are des…

超曲面非孤立奇点的不变量

Invariants of Non-Isolated Singularities of Hypersurfaces

本文推广了 Siersma、Pellikaan 和 de Jong 关于奇异轨迹为光滑曲线的超曲面 morsification 的结果，通过研究横截判别式及其与代数、拓扑不变量的关系，应用于 Yomdin 型孤立奇点研究。

On the quotient of Milnor and Tjurina numbers for two-dimensional isolated hypersurface singularities

In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

Corrigendum to "Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities"

We explicitly fix a mistake in a preliminary statement of our previous paper on the conductor at a multiplanar singularity. The correction is not immediate and, though the mistake does not affect correctness of the subsequent results, the wrong statement could easily be misleading.

delta不变量与孤立非正常奇点的同时规范化

The Delta Invariant and Simultaneous Normalization for Families of Isolated Non-Normal Singularities

本文研究了孤立非正常奇点的delta不变量及其在族中的上半连续性，提出在纤维上可规范化的条件，并推广了经典结果。

On the F-purity of isolated log canonical singularities

A singularity in characteristic zero is said to be of dense F-pure type if its modulo p reduction is locally F-split for infinitely many p. We prove that if $x \in X$ is an isolated log canonical singularity with $μ(x \in X) \le 2$ (see Definition 1.4 for the definition of the invariant $μ$), then it is of dense F-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense F-pure type in the case of three-dimensional isolated singularities.

Residues and duality for singularity categories of isolated Gorenstein singularities

We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.

Lyubeznik's invariants for cohomologically isolated singularities

In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This strengthens the results of Bondu and myself in math.AG/0406265 while at the same time somewhat simplifying the proofs. Through examples I further point out the bad behavior of these invariants under reduction to positive characteristic.

孤立超曲面奇点的F-纯阈值的界

Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities

本文在正特征代数闭域上，利用Milnor数和Tjurina数等经典奇点不变量，给出了孤立超曲面奇点F-纯阈值的界，并针对曲线奇点得到了基于值半群生成元的显式计算，推广了Igusa公式。

Maximal Modifications and Auslander-Reiten Duality for Non-isolated Singularities

We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modify…

Poisson structures over a complete intersection with isolated singularities

We study Poisson structures over singular varieties. In this purpose, we consider the Koszul complex associated to the equations of a complete intersection. This complex forms a differential graded algebra which is equivalent to the algebra of the variety. We show that a Poisson structure is equivalent to a sequence of multiderivations over the Koszul complex. If our variety has isolated singularities, then we can construct a sequence of multiderivations of reduced form.

On 1-forms on isolated complete intersection curve singularities

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $μ$ and the Tjurina number $τ$ of any isolated plane curve singularity satisfy the inequality $3μ<4τ$.