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 (

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.

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 will take place on Zoom but will also be recorded. Links to the videos will be posted here.

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.
Please send any comments/suggestions/errors to

The lectures and course notes draw from the following sources:

Stacks references: Moduli of curves references:
If you would like to participate informally in the class, please send me an email at 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.