Instructor: Andrew Kresch

Class meeting: MW 1-2, Room: 4N30

Course description:

Algebraic stacks are important in algebraic geometry because they play a role that ordinary schemes cannot in representing certain kinds of moduli problems. The first part of the course will be spent motivating and finally giving the definition of an algebraic stack. This much already involves quite of bit of terminology and theory (étale topology, descent, groupoids). The next part of the course will demonstrate some of the things we can do with stacks. For instance, once we have a theory of coherent sheaves it will be possible to give Deligne and Mumford's proof of the irreducibility of moduli space of curves in arbitrary characteristic. The last part of the course will be devoted to Artin stacks. In contrast to the stacks considered by Deligne and Mumford, which correspond to moduli problems in which the objects being parametrized have only finite automorphism groups, the more general kind of stacks known as Artin stacks allow infinite automorphism groups. Here the theory is technically harder, but the rewards are great. Moduli of vector bundles, and additional topics as time allows, will be covered.

Certain topics will be presented by outside speakers.

Prerequisites:

Some exposure to algebra and algebraic geometry. Important concepts (e.g., classes of morphisms such as étale, unramified, ...) will be reviewed as needed. It is imagined that the course will attract graduate students in the second year and beyond, as well as well-motivated first-year students. I hope some faculty members will decide to attend as well.

Text:

G. Laumon and L. Moret-Bailly, Champs Algébriques, Prépubl. math. de l'Université Paris-Sud, 1995. (photocopies will be available)

Course outline - click topic for lecture notes (as they become available)

• Motivation: indication of proof of irreducibility of moduli space (dvi / pdf) ; Picard groups of moduli problems (dvi / pdf)
• The étale topology on schemes (dvi / pdf)
• Faithfully flat descent (dvi / pdf)
• Categories fibered in groupoids (dvi / pdf) ; prestacks, stacks, and groupoid schemes (dvi / pdf)
• Algebraic stacks (dvi / pdf) ; properties of stacks and morphisms of stacks (dvi / pdf)
• Etale site of a Deligne-Mumford stack, sheaves and cohomology (dvi / pdf)
• Intersection theory (dvi / pdf) ; virtual fundamental class (dvi / pdf) ; sample calculation (dvi / pdf)
• Coarse moduli space (dvi / pdf) ; consequences of coarse moduli space (dvi / pdf)
• Orbifold fundamental group (dvi / pdf) ; fundamental groups of moduli spaces and a conjecture of Grothendieck (lecturer: David Harbater, dvi / pdf)
• Riemann-Roch theorem ; algebraization (lecturer: Bertrand Toen)
• Cohomology of Artin stacks (lecturer: Kai Behrend)

The following chapters of "the book" (K. Behrend, et al., Introduction to Stacks) are available in draft form.

• Intersection Theory
• Algebraic Stacks
• Appendix: Descent
• Bibliography