Dinamo Djounvouna

(Department of Mathematics, University of Manitoba)

Introduction to Moduli Spaces

Date Thursday, February 28, 2019

In algebraic geometry, classification is a key question. When studying geometric objects, it is desirable to classify them according to different criteria in order to be able to distinguish the equivalent classes in this category. Moduli problems are essentially classification problems. Given a collection of geometric objects, we want to classify them up to a notion of equivalence that we are given. Moduli spaces arise as spaces of solutions of geometric classification problems, and they may carry more geometric structures than the objects we are classifying. The construction of moduli spaces is important in algebraic geometry and difficult in general. To any moduli problem M, corresponds a moduli functor, and the study of the classification problem reduces to that of the representability of that functor. On the other hand, moduli spaces may arise as the quotient of a variety by a group action. Quotients of schemes by reductive groups arise in many situations. Many moduli spaces may be constructed by expressing them as quotients. Geometric Invariant Theory (GIT) gives a way of performing this task in reasonably general circumstances.

In this talk, I will introduce the notion of classification problem, moduli problem, fine moduli spaces and coarse moduli spaces in a functorial point of view.