Geometry of moduli spaces of algebraic varieties

模空间参数化代数变量或其他几何对象。它们是现代几何的基础，但就连它们的存在与否也常常是个难题。已知模空间存在的重要情况包括曲线和阿贝尔变分，它们彼此紧密相连，然后模空间本身具有几何结构，这是理解它们的手段。例如，已知这些模空间中的大多数是一般类型（即在大范围内复杂），但具有规范奇点（即在小范围内不太复杂）。然而，有许多有趣的特殊情况，这些情况往往是重要的。本研究的目的是研究其中一些特殊情况，并了解它们的几何形状。特别地，我将研究一些特殊的阿贝尔变体和它们上的泛族的模空间，并研究它们的奇异性和它们的整体几何。最近有一些一般性的结果，来自于Ma， Farkas和Verra， Scheithauer和Salvati Manni， Sacca和其他人，包括我的导师，这些结果告诉我们哪些病例是特殊的，可能会引起我们的兴趣。目标包括确定可能出现的奇点类型和计算双变量，如Kodaira维。在之前的工作中，将使用广泛的数学工具，包括模形式（来自数论）、表示理论（来自代数）以及几何方法。本研究属于纯数学领域，因此不能期望在项目的时间范围内对数学以外的领域产生直接影响。然而，代数几何是纯数学的一个主要部分，与数论和拓扑以及其他类型的几何相互作用，并且与理论物理密切相关，并且在计算机科学，密码学和许多其他领域具有特定的应用。它得到了EPSRC的大力支持：作为一个例子，我们提到了非常大的合作“分类，计算和构造：几何中的新方法”，但还有很多其他的。

Moduli spaces parametrise algebraic varieties or other geometric objects. They are fundamental to modern geometry but even their existence is often a hard question. Important cases where moduli spaces are known to exist include curves and abelian varieties, which are closely linked to one another, and then the moduli spaces have a geometric structure themselves, which is the means of understanding them. For example, it is known that most of these moduli spaces are of general type (that is, complicated on a large scale) but have canonical singularities (that is, not too complicated on a small scale). However, there are many interesting special cases, which are often important ones.The aim of this research is to examine some of those special cases and understand their geometry. In particular, I will examine some moduli spaces of special abelian varieties and universal families over them, and study their singularities and their global geometry. There are recent general results due to Ma, Farkas and Verra, Scheithauer and Salvati Manni, Sacca and others including my supervisor, and these tell us which cases are special and are likely to be of interest. The objectives include determining the types of singularities that can arise and computing birational invariants such as the Kodaira dimension. As in this previous work, a wide range of mathematical tools will be used, including modular forms (from number theory) an representation theory (from algebra) as well as geometric methods.This research is in pure mathematics and therefore cannot be expected to have direct impact outside mathematics within the timescale of the project. Algebraic geometry, however, is a major part of pure mathematics, interacting with number theory and topology as well as other kinds of geometry, and is closely linked to theoretical physics as well as having specific applications in computer science, cryptography and many other areas. It has been heavily supported by EPSRC: as an example we mention the very large collaboration "Classification, Computation and Construction: New Methods in Geometry", but there are many others.