Math 582C: Introduction to stacks and moduli
Winter quarter 2021, University of Washington
Lectures: Mon/Wed 11:3012:50 (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 DeligneMumford stack of dimension \(3g3\) 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): Introductionmotivation, history and the functorial worldview
 Lecture 2 (Jan 6): The etale experiencesites 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
 Jan 18: no class (Martin Luther King Jr. Day)
 Lecture 5 (Jan 20): Algebraic spaces and stacks
 Covers definitions of algebraic spaces, DeligneMumford stacks and algebraic stacks (2.1).
 Summarized effective descent (App B.3) and existence of Hilbert schemes (App D).
 video, slides
 Jan 25: no class
 Lecture 6 (Jan 27): First properties
 First properties of algebraic spaces and stacks (2.2)
 Equivalence relations and groupoids of schemes (2.3)
 video, slides
 Lecture 7 (Feb 1): Representability of the diagonal
 Covered pathological examples of algebraic spaces (some of 2.9)
 Representability of the diagonal (2.4)
 video, slides
 Lecture 8 (Feb 3): Dimension, tangent spaces, and residual gerbes
 Lecture 9 (Feb 8): Characterization of DeligneMumford stacks
 Characterization of DeligneMumford stacks (2.6)
 Smoothness and the formal lifting criterion (2.7)
 Properness and the valuative criterion (2.8)
 Upshot: we now know that \( \mathcal{M}_g \) is a DeligneMumford stack smooth over Spec \(\mathbb{Z}\) of relative dimension 3g3.
 video, slides
 Lecture 10 (Feb 10): Geometry of DeligneMumford stacks
 Recalled and elaborated on separated and properness, and their valuative criteria (2.8)
 Quasicoherent sheaves on DeligneMumford stacks (3.1)
 Local structure theorem for DeligneMumford stacks (3.2)
 video, slides
 Feb 15: no class (President's Day)
 Lecture 11 (Feb 17): Existence of coarse moduli spaces
 Lecture 12 (Feb 22): Nodal curves
 Recapped the KeelMori theorem (3.3)
 Nodal curves (4.1)
 video, slides
 Lecture 13 (Feb 24): Stable curves
 Covered pointed stable curves (4.2): definition, equivalent characterizations, wellbehaved with respect to pointed normalization, positivity of dualizing sheaf, families of stable curves, openness of stability, characterization of automorphisms, deformations and obstrctions
 video, slides
 Lecture 14 (Mar 1): Stack of all curves
 Summarized the six steps toward projective moduli (0.9)
 Covered the contraction of rational tails and bridges in families of prestable curves (end of 4.2)
 Showed that the stack of all curves is algebraic and concluded that \( \bar{\mathcal{M}}_{g,n} \) is algebraic (4.3)
 video, slides
 Lecture 15 (Mar 3): Stable reduction
 Summarized where we are: \( \bar{\mathcal{M}}_{g,n} \) is a quasicompact DeligneMumford stack smooth over Spec \(\mathbb{Z}\) of relative dimension 3g3+n
 Gave first examples of stable reduction for colliding marked points and a nodal family degenerating to a cusp
 Gave general strategy to establish stable reduction based on the birational geometry of surfaces (and specifically Embedded Resolutions). Gave proof in characteristic 0. (4.4)
 video, slides
 Lecture 16 (Mar 8): Explicit stable reduction and gluing & forgetful morphisms
 Computed stable reduction of a degeneration y^2=x^5+t (4.4)
 Showed the uniqueness of the stable limit, i.e. separatedness of \( \bar{\mathcal{M}}_{g,n} \) (4.4)
 Defined the gluing maps (4.5)
 \( \bar{\mathcal{M}}_{i,n} \times \bar{\mathcal{M}}_{gi,m} \to \bar{\mathcal{M}}_{g,n+m}\)
 \( \bar{\mathcal{M}}_{g,n} \to \bar{\mathcal{M}}_{g+1,n2}\)
 Showed that the forgetful map \( \bar{\mathcal{M}}_{g,1} \to \bar{\mathcal{M}}_{g} \) is a universal family (4.5)
 video, slides
 Lecture 17 (Mar 10): Irreducibility
 Covers 4.6 in notes.
 Background on branched covers
 ClebschHurwitz argument for connectedness of \(M_g\) over \(\mathbb{C}\) (1872 & 1891)
 Fulton's (completely algebraic) argument for irreducibility of \(\bar{M}_g\) in characteristic 0, which is in an appendix to Harris & Mumford's paper "On the Kodaira Dimension of the Moduli Space of Curves." (1982)
 The irreducibility arguments by DeligneMumford and Fulton in 1969 which show irreducibility of \(M_g\) in characteristic p (with \(p > g+1\) in Fulton's paper)
 DeligneMumford's two arguments in "On the irreducibility of space curves of given curves" (1969)
 Fulton's argument in "Hurwitz schemes and Irreducibility of Moduli of Algebraic Curves" (1969)
 video, slides
 Lecture 18 (Mar 15): Projectivity
 Covers 4.7 in notes.
 Recap of how we got here
 Setup for \(\bar{M}_g\)
 Survey of projectivity methods: (a) GIT, (b) Hodge theory, (c) positivity of line bundles
 Background on nef vector bundles
 Kollár's ampleness lemma in "Projectivity of Complete Moduli" (proof uses NakaiMoishezon)
 Application to \(\bar{M}_g\)
 Thm 1: Nefness of \(\pi_*(\omega_{\mathcal{C}/T}^{\otimes k} ) \) for a family of stable curves \( \pi: \mathcal{C} \to T\) over a curve for \(k \gg 0\) implies that \(\lambda_k = \det \pi_*(\omega_{\mathcal{U}_g/\bar{\mathcal{M}}_g}^{\otimes k} ) \) is ample on \( \bar{M}_g \) for \(k \gg 0\). (Proof uses ampleness lemma)
 Thm 2: For a family of stable curves \( \pi: \mathcal{C} \to T\) over a curve, \(\pi_*(\omega_{\mathcal{C}/T}^{\otimes k} ) \) is nef for \( k \ge 2\). (Proof uses reduction to char=p and Ekedahl's theorem on the vanishing of \(H^1(S, \omega_S^{\otimes (n)})\) for \(n \ge 1\) and for smooth proj minimal surfaces \(S \) of general type)
 Thm 1 & 2 imply \(\bar{M}_g\) is projective!!!
 video, slides
 Spring break!
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 jarod@uw.edu.
The lectures and course notes draw from the following sources:
Stacks references:
 Champs algébriques, Laumon and MoretBailly
 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, HalpernLeistner
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.