Course Title
An Introduction to Moduli Spaces and Stacks
Abstract
The lectures will give a compact introduction to moduli spaces and stacks, centered on the moduli of curves. We will begin with moduli functors, families, fine moduli spaces, and Hilbert schemes as a fundamental representability result. We will then explain why many natural moduli problems cannot be represented by schemes, focusing on isomorphisms, automorphisms, and the passage from embedded objects to objects up to equivalence. This leads to quotient stacks and the basic ideas of algebraic and Deligne–Mumford stacks. The final part of the course will focus on the moduli stack \(\mathcal{M}_g\) of smooth curves and the compactification \(\overline{\mathcal{M}}_g\) of stable curves.
Lecture Topics
- Moduli Functors, Grassmannians, and Hilbert Schemes.
- From Quotients to Stacks: Automorphisms and Quotient Stacks.
- The Moduli Stack of Smooth Curves \(\mathcal{M}_g\).
- Stable Curves and the Compactification \(\overline{\mathcal{M}}_g\).
Problem Sets & Materials
Lecture 1 Problems: Moduli Functors, Grassmannians, and Hilbert Schemes (PDF)
Lecture 2 Problems: Automorphisms, Torsors, and Quotient Stacks (PDF)
Lecture 3 Problems: Algebraic and Deligne–Mumford Stacks, Moduli of Curves (PDF)