## 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
• Jan 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).
• 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 Deligne-Mumford stacks
• Characterization of Deligne-Mumford 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 Deligne-Mumford stack smooth over Spec $$\mathbb{Z}$$ of relative dimension 3g-3.
• video, slides
• Lecture 10 (Feb 10): Geometry of Deligne-Mumford stacks
• Recalled and elaborated on separated and properness, and their valuative criteria (2.8)
• Quasi-coherent sheaves on Deligne-Mumford stacks (3.1)
• Local structure theorem for Deligne-Mumford 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 Keel-Mori theorem (3.3)
• Nodal curves (4.1)
• video, slides
• Lecture 13 (Feb 24): Stable curves
• Covered pointed stable curves (4.2): definition, equivalent characterizations, well-behaved 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 quasi-compact Deligne-Mumford stack smooth over Spec $$\mathbb{Z}$$ of relative dimension 3g-3+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)
• $$\bar{\mathcal{M}}_{i,n} \times \bar{\mathcal{M}}_{g-i,m} \to \bar{\mathcal{M}}_{g,n+m}$$
• $$\bar{\mathcal{M}}_{g,n} \to \bar{\mathcal{M}}_{g+1,n-2}$$
• 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
• Clebsch-Hurwitz 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 Deligne-Mumford and Fulton in 1969 which show irreducibility of $$M_g$$ in characteristic p (with $$p > g+1$$ in Fulton's paper)
• Deligne-Mumford'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 Nakai-Moishezon)
• 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.

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.