Unfoldings of maps, the first results on stable maps, and results of Mather-Yau/Gaffney-Hauser type in arbitrary characteristic

局部Euler阻塞与行列式簇的Chern-Mather类

Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties

对于$m\geq n$，设$K$是一个代数闭基域，定义$τ_{m,n,k}$为$K$上具有核维数$\geq k$的$m\times n$矩阵的集合.这是$\mathbb{P}^{mn-1}$的投射子簇，通常称为行列式簇。在大多数情况下，$τ_{m,n,k}$是单数，具有单数位点$τ_{m,n,k+1}$。本文计算了$τ_{m,n,k}$的局部Euler阻塞，证明了$τ_{m,n,k}$的交上同调复形的特征环是不可约的.我们还给出了$τ_{m,n,k}$的Chern-Mather类作为射影空间中的类的一个显式公式。不可约的相交上同调特征循环如下从明确计算的本地欧拉障碍，研究的'Tjurina变换'的行列式品种，和柏原-Dubson的microlocal指数定理。我们的显式公式是基于Grassmannian上的万有丛的某些陈类的度的计算。我们使用Macaulay 2中的Schubert 2包展示了对于$m,n,k$的一些小值的Chern-Mather类和$τ_{m,n,k}$的特征循环类的例子。在复数上，$τ_{m,n,k}$的局部欧拉阻塞最近由N.~ Grulha，T.~加夫尼和M.~在复杂的几何方法Ruas。

Deformations of corank $1$ frontals

We develop a Thom-Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather-Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with …

任意特征下映射的左右等变轨道

Orbits of the left-right equivalence of maps in arbitrary characteristic

本文填补了纯代数方法在研究映射左右等变群轨道上的空白，建立了任意特征下从切空间到群轨道的一般过渡准则，并发展了左右群切空间的混合模结构。