Geometry of moduli spaces of algebraic varieties

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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.