Syllabus:
This course will cover algebraic curves and their moduli. Topics will include:
- Smooth curves: Riemann-Roch, Serre-Duality, Riemann-Hurwitz, positivity of line bundles on curves, families of smooth curves.
- Examples of curves: classification of curves of low genus, hyperelliptic curves
- Nodal curves: nodal singularities and their characterizations, genus formula for nodal curves, the dualizing sheaf \(\omega_C\), families of nodal curves, local structure of nodes
- Stable curves: definitions and equivalences, positivity of \(\omega_C\), families of stable curves, deformation theory, stabilization.
- Detour on stacks and moduli: overview of what a moduli space is, streamlined intro to sites, stacks, and algebraic stacks, algebraicity of \(\mathcal{M}_g\) as a quotient of Hilbert scheme
Possible additional topics:
- Algebraicity of the stack of all curves
- Stable reduction: proof in characteristic zero, explicit computing the stable reduction in examples, separated and properness of \(\overline{\mathcal{M}}_g\)
- Gluing and forgetful morphisms: boundary divisors and description of the universal family of \(\overline{\mathcal{M}}_g\) as \(\overline{\mathcal{M}}_{g,1}\)
- Irreducibility of \(\mathcal{M}_g\)
- Projectivity of the coarse moduli space \(\overline{M}_g\).
- Geometry of \(\overline{\mathcal{M}}_g\): picard groups, Chow groups for small \(g\), birational geometry for small \(g\), Harris-Mumford's theorem on the general typeness of \(\overline{\mathcal{M}}_g\)
Prerequisites:
Introductory algebraic geometry. We will use the language of schemes but it is possible to understand much of the theory (but not all) using the language of varieties.
References:
The main reference for this course is Chapter 5 `Moduli of Stable Curves' of my book-in-progress
Stacks and Moduli.
In the Winter of 2021, I taught a similar course
Math 582C: Introduction to Stacks and Moduli. Videos and slides of the lecture notes are available on the course website. This course covered much more material on the foundations of algebraic stacks than we will cover this quarter, and the pace of the lectures went faster than we will go.
The book-in-progress `Stacks and Moduli' contains extensive references to the literature with pointers to the original references and other expositions. The main recommended references for the material on curves are:
- The irreducibility of the space of curves of given genus, Deligne and Mumford
- Moduli of Curves, Harris and Morrison
- The Stacks Project
Homework:
- Complete exercises in the textbook.
- Find as many typos and errors in the book as you can.