拟常映射临界轨迹的丛分解

On the Milnor fiber boundary of a quasi-ordinary surface

We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. We indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.

Congruence relations for Shimura varieties associated to unitary groups

We prove the congruence relation for the mod-p reduction of Shimura varieties associated to a unitary similitude group GU(n-1,1), when p is inert and n odd. When n is even, this result was obtained by T. Wedhorn and O. Bültel using the density of the ordinary locus in the p-isogeny space. This condition fails in our case. A key element is the understanding of the supersingular locus, for which we refer to the two articles by I. Vollaard and T. Wedhorn. The proof makes extensive use of elementary algebraic geometry, but also some deeper results.

Jet schemes of quasi-ordinary surface singularities

In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity. We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on $\mathbb{A}^3$), that computes the log canonical threshold of the singularity embedded in $\mathbb{A}^3$.…

On birational superrigidity and conditional birational superrigidity of certain Fano hypersurfaces

We prove birational superrigidity of every hypersurface of degree N in P^N with singular locus of dimension s, under the assumption that N is at least 2s+8 and it has only quadratic singularities of rank at least N-s. Combined with the results of I. A. Chel'tsov and T. de Fernex, this completes the list of birationally superrigid singular hypersurfaces with only ordinary double points except in dimension 4 and 6. Further we impose an additional condition on the base locus of a birational map to a Mori fiber space. Then we prove conditional birational superrigidity of certain smooth Fano hypers…

拟普通奇点、本质除子与Poincaré级数

Quasi Ordinary Singularities, Essential Divisors and Poincare Series

本文通过单项赋值序列定义了拟超曲面奇点的Poincaré级数，证明其为有理函数及解析不变量，并证明其与特征单项式规范序列相互确定，从而刻画了奇点的嵌入拓扑类型。

Embeddings of the complex ball into Siegel space

We study properties of a map from a certain unitary group in $n$ variables to a related unitary group in $\binom{n}{k}$ variables. We explain how it gives rise to a map between canonical models of Shimura varieties and we prove that it extends to the ordinary locus of the integral model. Finally we extend results of Satake on the endomorphism ring of a generic image point to positive characteristics.

PEL型Shimura品种约化中的Kottwitz-Rapoport和p-秩层

Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type

本文研究了具有Iwahori水平结构的PEL型Shimura簇的某些积分模型的约化。在这些空间上，我们有Kottwitz-Rapoport和$p$秩分层。我们证明了KR层上的$p$ -秩是常数，推广了Ngô和Genestier的一个结果.我们证明了一个抽象的，统一的公式的$p$ -秩的KR层。在辛和在单一的情况下，我们得到明确的公式，其价值。我们将这些公式应用于普通轨迹的密度问题和$p$秩为0的轨迹的维数问题。

特征为2的秩2向量丛上Frobenius映射的作用

The action of the Frobenius map on rank 2 vector bundles in characteristic 2

设$X$是定义在特征为2的代数闭域上的普通平滑曲线。绝对Frobenius在秩为2的具有固定平凡行列式的向量丛的模空间$M_X$上诱导出一个有理映射$F$.如果$X$的亏格是2，则模空间$M_X$同构于3维的射影空间（如在复数上）。在这种情况下，我们明确给出$F$的方程，这使我们能够确定，例如，它的基本轨迹（一个点）和它的图像（不同于$M_X$）。

避免Veronese的超曲面

Veronese Avoiding Hypersurfaces

本文研究避免Veronese的超曲面，通过Macaulay逆系统重新解释平滑情况下的条件，并在奇点情况下证明减少的超曲面在特定奇点条件下为避免Veronese的超曲面，同时分类了奇异平面三次曲线。