Stacks
In this reading group, we classify geometric objects up to isomorphism, and how they vary in families. Examples include smooth projective algebraic curves, vector bundles, principal bundles, and stable maps. To do this, we construct moduli spaces of these objects as stacks rather than schemes, varieties, manifolds, or other topological spaces.
Though powerful for solving classification problems, stacks may seem unintuitive and technical at first. To demystify them, we think of stacks geometrically much like manifolds or schemes, they have dimension, smoothness, tangent spaces at points, symplectic structures, and compactifications. Important constructions include quotient stacks, algebraic stacks, and Deligne-Mumford stacks.
Our main source is Stacks and Moduli by Jarod Alper.
Organizer: Emanuel Roth.
Schedule:
- 14/1/24 Motivating stacks, moduli problems, Emanuel Roth
- 21/1/24 Defining stacks, Grothendieck topologies, e.g. étale and fppf topology, Ander Martin Iribar
- 28/1/24 Algebraic stacks and quotient stacks, Tudor Caba
- 4/2/24 Moduli stacks of bundles and curves, dimension and tangent spaces of stacks, Yaoqi Yang
- 11/2/24 Deformation theory of stacks, cotangent complexes, Shing Tak Lam
- 18/2/24 Quasicoherent sheaves of Deligne-Mumford stacks, Keel-Mori theorem, Emanuel Roth
- 25/2/24 Stacks in enumerative geometry, obstruction theory of Deligne-Mumford stacks, Noah Dizep
- 4/3/24 Stacks and stable homotopy theory, the stack of formal group laws, Tudor Caba
- 11/3/24 Good moduli spaces, theta-stratifications, S-completeness, Karim Réga