Birational geometry of higher dimensional varieties

拟议的研究涉及高维复代数几何的主题（即研究多个复变量的多项式方程组的解）。两个品种（即不可约解集）如果曲面有同构的开子集，则曲面是双有理的。该项目的主要重点是复射影变量的双有理几何中的自然问题，特别是与最小模型程序有关的问题。曲面的双有理分类在二十世纪初由意大利学派理解。如果曲面不被有理曲线覆盖，则存在是一个自然选择的双有理曲面（称为最小模型），它有许多有用的性质。最小模型程序旨在将这些结果推广到更高的维度。这个程序在第三维度上是完整的（由Mori，Kawamata，Kollar，Reid，Shokurov等人完成），并且已知适用于一般类型的品种（由Birkar，Cascini，Hacon，McKernan，Siu等人完成）。这个项目希望回答一些在这种情况下自然出现的遗留问题和疑问。

The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry (i.e. the study of the solutions of sets of polynomial equations in several complex variables). Two varieties (i.e. irreducible sets of solutions) are birational if they have isomorphic open subsets.The main focus of this project is on natural questions inthe birational geometry of complex projective varietiesand in particular on problems related to the Minimal Model Program.The birational classification of surfaces was understood by theItalian school at the beginning of the twentieth century.If a surface is not covered by rational curves then there is a naturalchoice 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 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.