Preliminary meeting:Thursday February 17th, Aula M2 (or email me to connect by teams).
The course will then be held in a weekly basis in the period March-May.
Preliminary schedule:thursdays at 16:00, room M1 (or at teams) The aim of the
course is to give an introduction to the theory of algebraic stacks,
focusing on its application to moduli theory.
We start by giving motivation and an overview of the program, which will follow mostly Jarod Alper’s Notes on Stacks and moduli.
According to the interests of participants, further topics can be covered in the last part of the course.
The course will be mostly covered by lectures from participants, according to the program and references for each lecture.
1. Program •
Motivation and overview of the program. Guiding examples: the moduli
spaces of curves and the moduli spaces of vector bundles on a curve. • Basics on category theory. Sheaves and Grothendieck topologies. • Groupoids and pre-stacks. • Stacks • Algebraic spaces and algebraic stacks • Geometric properties of algebraic stacks • Characterization of Deligne-Mumford stacks • Geometry of Deligne-Mumford stacks • Existence of coarse and good moduli spaces • The moduli stack of curves • Stacks on different geometric contexts (topological stacks, cone stacks, etc)
Meetings
(notes are available at teams) • 1. (17/02/2022): program and outline • 2. (03/03/2022): Background and motivation. Moduli functors. Examples. (Margarida) • 3. (10/03/2022): Sites and Grothendieck topologies. (Ismaele Vanni, Sapienza). • 4. (17/03/2022): Pseudofunctors and Prestacks. (Ismaele Vanni, Sapienza) • 5. (24/03/2022):Stacks, descent and examples, I. (Davide Gori, Sapienza) • 6. (31/03/2022): Stacks, descent and examples, II. (Davide Gori, Sapienza) • 7. (07/04/2022): Hilbert and Quot schemes. (Simon Schirren, Roma Tre) • 8. (21/04/2022): Algebraic Stacks. (Roberto Vacca, Tor Vergata) • 9. (05/05/2022): Algebraic Stacks II and Deligne Mumford stacks I. (Roberto Vacca and Nelson Alvarado, Tor Vergata) • 10. (11/05/2022): Deligne Mumford stacks II. (Nelson Alvarado, Tor Vergata) • 11. (18/05/2022): Equivalence Relations and groupoids of schemes. (Luca Casarin, Sapienza)