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Room P3.10, Mathematics Building

Barbara Fantechi, SISSA

An introduction to algebraic stacks

  1. Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.
  2. Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).
  3. More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.