Birational geometry of algebraic varieties

建议的研究涉及高维代数几何的主题。它主要集中在射影簇的二元几何中的自然问题，如与最小模型规划有关的问题的研究。PI将研究非一般类型簇的极小模型的存在性、丰度和翻转的终止、正特征中三重翻转的存在、Fano簇的几何(特别是LOG-FANO的有界性)、正规簇和乘子理想层的奇点、代数簇的多元映射和不规则簇的二元几何。曲面的二元分类在20世纪初被意大利学派所理解。如果曲面没有被有理曲线覆盖，则自然会选择双曲面(称为极小模型)，它具有许多有用的性质。最小模型计划旨在将这些结果推广到更高的维度。这个程序在3维空间上是完全的(由Mori，Kawamata，Kollar，Reid，Shokurov等人完成)，并且已知用于一般类型的复射影变种(由Birkar，Cascini，Hacon，McKernan，Siu等人工作)。这个项目希望回答在这一背景下自然产生的一些剩余的问题和猜测。

The proposed research deals with topics in Higher dimensional Algebraic Geometry. It is mainly focused on natural questions in the birational geometry of projective varieties such as the study of questions related to the minimal model program. The PI will investigate the existence of minimal models for varieties not of general type, abundance and the termination of flips, the existence of flips for 3-folds in positive characteristic, the geometry of Fano varieties (in particular the boundedness of log-Fanos), the singularities of normal varieties and multiplier ideal sheaves, the pluricanonical maps of algebraic varieties and the birational geometry of irregular varieties.The birational classification of surfaces was understood by the Italian school at the beginning of the twentieth century. If a surface is not covered by rational curves then there is a natural choice of a birational surface (known as the minimal model) which has many useful properties. The Minimal Model Program aims to generalize these results to higher dimensions. This program is complete in dimension 3 (by work of Mori, Kawamata, Kollar, Reid, Shokurov and others) and is known to work for complex projective varieties of general type (by work of Birkar, Cascini, Hacon, McKernan, Siu and others). This project hopes to answer some of the remaining questions and conjectures that naturally arise in this context.