Geometry of moduli spaces of algebraic varieties

代数簇模空间的几何

• 批准号: 2445863
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2445863
• 关键词: Geometry; moduli; spaces; algebraic; varieties

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.

模空间参数化代数簇或其他几何对象。它们是现代几何学的基础，但甚至它们的存在也常常是一个难题。已知模空间存在的重要情况包括曲线和阿贝尔簇，它们彼此紧密相连，然后模空间本身具有几何结构，这是理解它们的方法。例如，已知这些模空间大多数是一般类型的（也就是说，在大尺度上是复杂的），但具有正则奇点（也就是说，在小尺度上不太复杂）。然而，有许多有趣的特殊情况，这往往是重要的。本研究的目的是检查其中一些特殊情况，并了解他们的几何。特别是，我将研究一些模空间的特殊交换品种和泛家庭，并研究他们的奇异性和他们的整体几何。最近，由于Ma、Farkas和Verra、Scheithauer和Salvati Manni、Sacca以及包括我的导师在内的其他人的研究，我们得到了一些一般性的结果，这些结果告诉我们哪些情况是特殊的，哪些情况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有理不变量，如科代拉维。与之前的工作一样，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。这项研究是纯数学的，因此不能期望在项目的时间范围内对数学之外产生直接影响。然而，代数几何是纯数学的一个主要部分，与数论和拓扑学以及其他类型的几何相互作用，与理论物理密切相关，并在计算机科学，密码学和许多其他领域有具体的应用。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：几何学的新方法”，但还有许多其他的。