Families of varieties of general type

PI将研究曲面及其变化。从计算的角度来看，最好的曲面是那些可以用多项式方程描述的曲面。并不是每个曲面都是这样的，但纳什证明了每个曲面都可以被这样的多项式定义的曲面逼近。该建议的主要目的是了解如果我们改变定义多项式方程的系数，曲面是如何变化的。PI尤其旨在了解曲面族退化为非常复杂、高度奇异的曲面时的情况。其主要目的是开发可用于简化此类奇点的方法。作为该项目的一部分，PI还旨在研究曲面上的曲线族，特别是曲面奇点附近的曲线族。PI旨在研究一般类型的代数曲面族。对于所有的族，都有一个通用空间，称为模空间。这个模空间不是紧的，PI想要证明，通过考虑具有半对数标准奇点和充足标准类的曲面，可以给出一个好的紧化。PI的目的是在更高的维度上发展类似的理论，在这里，各种一般类型的规范模型提供了基本的对象。作为这个项目的一部分，PI旨在了解半对数正则奇点的结构，特别是它们的分解的组合结构。PI旨在了解决议的对偶复合体的结构。一个密切相关但原则上独立的项目是通过奇点了解弧线的纳什空间的结构。最初的纳什猜想在3维及以上维度上被证明是错误的，但有一个考虑到这些例子的修改。PI旨在研究一些非常具体的例子和一些普遍的现象。

The PI will study surfaces and their variations. From the computational point of view, the best surfaces are those that can be described by a polynomial equation. Not every surface is such but Nash proved that every surface can be approximated by such polynomially defined surfaces. The main aim of the proposal is to understand how surfaces vary if we change the coefficients of the defining polynomial equations. The PI would especially aim to understand situations when a family of surfaces degenerates to a very complicated, highly singular surface. The main aim is to develop methods that can be used to simplify such singularities. As part of this project, the PI also aims to study families of curves on surfaces, especially near the singular points of the surface.The PI aims to study families of algebraic surfaces of general type. There is a universal space for all families, called the moduli space. This moduli space is not compact, the PI intends to prove that a good compactification is given by considering surfaces that have semi-log canonical singularities and ample canonical class. The PI aims to develop a similar theory in higher dimensions, here the canonical models of varieties of general type provide the basic objects. As part of this project, the PI aims to understand the structure of semi-log canonical singularities, especially the combinatorial structure of their resolution. The PI aims to understand the structure of the dual complex of the resolution. A closely related but in principle independent project is to understand the structure of the Nash space of arcs through a singularity. The original conjectures of Nash were disproved in dimensions 3 and up, but there is a modification that takes these example into account. The PI aims to study both some very concrete examples and some general phenomena.