Infinitesimal Automorphisms of Algebraic Varieties

本课题的目的是研究具有正特征的光滑射影簇的自同构方案。在这一背景下，无穷小自同构，即包含在单位的无穷小邻域中的自同构方案的子群方案，起着基本的作用。这与特征0中的情况形成了鲜明对比，在特征0中，这样的非缩减组方案甚至不存在。这个项目的第一部分将是在特殊类型的非直纹曲面的情况下的无穷小自同构的显式计算，以及利用这些计算的结果构造高维种类，如Calabi-Yau三重曲面。在第二部分中，我想从更抽象的角度研究曲面和一般类型簇的自同构群方案及其典范模型的方案结构，并努力给出这种自同构方案的长度的界。

The purpose of this project is to study the automorphism scheme of smooth projective varieties in positive characteristic. In this context, infinitesimal automorphisms, i.e. subgroup schemes of the automorphism scheme which are contained in an infinitesimal neighborhood of the identity, play a fundamental role. This is in stark contrast to the situation in characteristic 0 where such non-reduced group schemes do not even exist. The first part of this project will be the explicit calculation of infinitesimal automorphisms in the case of non-ruled surfaces of special type and the construction of varieties of higher dimension, such as Calabi-Yau threefolds, using the results of these computations. In the second part, I want to study the scheme structure of the group scheme of automorphisms of surfaces and varieties of general type along with their canonical models from a more abstract perspective and work towards giving a bound on the length of this automorphism scheme.