Oberwolfach-Seminar: Algebraic Stacks

Date:

October 15th - October 21st, 2006

Organizers:

Kai Behrend, Vancouver

William Fulton, Michigan

Andrew Kresch, Warwick

Programme:

Grothendieck's insights paved the way for stacks to emerge as important objects in algebraic geometry and other disciplines. Stacks often capture the geometry of a moduli problem better than traditional algebraic varieties, or even schemes. Deligne and Mumford identified a class of stacks, which they called `algebraic', over which many of the constructions and results of algebraic geometry are applicable. In an important paper in 1969, they proved the irreducibility of the moduli space of smooth algebraic curves of given genus, over an arbitrary base field, and by introducing stacks they managed to give a conceptually simple and very appealing proof. Later Artin identified a more general class of algebraic stacks and gave a general criterion for a given moduli problem to correspond faithfully to such a stack. In the more than 30 years that have passed since these early foundational works appeared, stacks have had many important applications, for instance to the foundations of Gromov-Witten theory and to the study of Brauer groups. Stacks are now an indispensable tool to the modern algebraic geometer, and are important in related fields such as differential geometry (where they are known as orbifolds, or in older papers, V-manifolds) and mathematical physics. This seminar will give an introduction to the foundations of stacks as well as a taste of several applications. Examples will be emphasized, and participants will obtain practice in computing invariants of stacks, such as the Picard group, and will see how stacks can be used as a tool in the study of algebraic varieties. Specific topics to be studied include: definition of Deligne-Mumford stack, moduli stacks, irreducibility proof of Deligne and Mumford, stacky curves, results on Brauer groups.

Prerequisites:

Some knowledge of algebraic geometry, e.g., Chapters I and II of R. Hartshorne, Algebraic Geometry

Deadline for applications:

September 1, 2006

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany
.