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We are going to study stacks while keeping moduli spaces in mind. In particular, we want to understand the following theorem:
Theorem: 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.
We shall follow Jarod Alper's lectures from his course on stacks in Spring 2021. Here is a link to his course website.
Resources
Here is a list of resources, lovingly stolen from Jarrod's website.
Jarods' lecture notes: Course notes and another set of slightly reordered notes
Stacks references:
Algebraic spaces, Knutson
Notes on Grothendieck topologies, fibered categories and descent theory, Vistoli
Algebraic spaces and stacks, Olsson
Champs algébriques, Laumon and Moret-Bailly
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
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