Introduction to Stacks and Moduli Lecture notes for Math 582C, working draft University of Washington, Winter 2021 Jarod Alper
[email protected] February 1, 2021 2 Abstract These notes provide the foundations of moduli theory in algebraic geometry using the language of algebraic stacks with the goal of providing a self-contained proof of the following theorem: Theorem A. The moduli space Mg of stable curves of genus g ≥ 2 is a smooth, proper and irreducible Deligne{Mumford stack of dimension 3g − 3 which admits a projective coarse moduli space.1 Along the way we develop the foundations of algebraic spaces and stacks, and we hope to convey that this provides a convenient language to establish geometric properties of moduli spaces. Introducing these foundations requires developing several themes at the same time including: • using the functorial and groupoid perspective in algebraic geometry: we will introduce the new algebro-geometric structures of algebraic spaces and stacks; • replacing the Zariski topology on a scheme with the ´etaletopology: we will generalize the concept of a topological space to Grothendieck topologies and systematically using descent theory for ´etalemorphisms; and • relying on several advanced topics not seen in a first algebraic geometry course: properties of flat, ´etaleand smooth morphisms of schemes, algebraic groups and their actions, deformation theory, Artin approximation, existence of Hilbert schemes, and some deep results in birational geometry of surfaces. Choosing a linear order in presenting the foundations is no easy task. We attempt to mitigate this challenge by relegating much of the background to appendices. We keep the main body of the notes always focused entirely on developing moduli theory with the above goal in mind.