## Math 582C: Introduction to stacks and moduli

#### Winter quarter 2021, University of Washington

Lectures: Mon/Wed 11:30-12:50 (beginning Mon Jan 4, 2021)
Instructor: Jarod Alper (jarod@uw.edu)

Syllabus: The goal of this course is to establish the following theorem:

The moduli space $$\bar{\mathcal{M}}_g$$ of stable curves of genus $$g\ge2$$ is a smooth, proper and irreducible Deligne-Mumford stack of dimension $$3g-3$$ which admits a projective coarse moduli space.

Along the way we will develop the foundations of algebraic spaces and stacks, and in particular we will precisely define each term in the above theorem.

Prerequisites:
You should have some prior exposure to scheme theory and a willingness to accept on faith a handful of results (e.g. existence of Hilbert schemes, Artin approximation, resolution of singularities of surfaces, some deformation theory, ...), some of which might be difficult to prove but at least the statements and their applications are easy to internalize.

Lectures:
Lectures will take place on Zoom but will also be recorded. Links to the videos will be posted here.
• Lecture 1 (Jan 4): Introduction--motivation, history and the functorial worldview
• Lecture 2 (Jan 6): The etale experience--sites and sheaves
• Lecture 3 (Jan 11): Groupoids and prestacks
• Lecture 4 (Jan 13): Stacks
• Covers fibered products of prestacks (end of 1.3) and stacks (1.4)
• video, slides
• January 18: no class (Martin Luther King Jr. Day)
• Lecture 5 (Jan 20): Algebraic spaces and stacks
• Covers definitions of algebraic spaces, Deligne-Mumford stacks and algebraic stacks (2.1). Also summarized effective descent (App B.3) and existence of Hilbert schemes (App D).
• video, slides
• January 25: no class

Online course notes:
Below is a working draft of the lecture notes which will be continually edited and expanded during the course. The notes include a long introduction containing motivation for the theory of moduli stacks and how it can be used to construct projective moduli spaces. The course however will only quickly cover this motivation. If this content is unfamiliar to you, you may want to read the intro as background for the lectures.

The lectures and course notes draw from the following sources:

Stacks references:
• Champs algébriques, Laumon and Moret-Bailly
• Algebraic spaces, Knutson
• Notes on Grothendieck topologies, fibered categories and descent theory, Vistoli
• Algebraic spaces and stacks, Olsson
• The Stacks Project
• Lecture notes on moduli theory, Halpern-Leistner
Moduli of curves references:
• The irreducibility of the space of curves of given genus, Deligne and Mumford
• Moduli of Curves, Harris and Morrison
• Projectivity of complete moduli, Kollár
• The Stacks Project

Participation:
If you would like to participate informally in the class, please send me an email at jarod@uw.edu with (1) your name, (2) email address, (3) affiliation (if any), (4) status (e.g. 3rd yr PhD student, postdoc, ...), and (5) a one sentence summary of your background in algebraic geometry.