Jochen Heinloth: The Moduli Stack of Vector Bundles on a Curve

We will give an introduction to algebraic stacks, using only basic algebraic geometry. Our main example will be the moduli stack of vector bundles on a projective algebraic curve, which provides a lot of interesting phenomena.
To illustrate the usefulness of the concept, we will compare it to the classical approach using coarse moduli schemes. Here we will see that questions like the computation of cohomological invariants can already be formulated for the stack and tend to become much simpler in this setting. Moreover, these results can be used to deduce information on the coarse moduli space.

Literature:
To get some impression of what stacks are, there are several introductions to algebraic stacks available, e.g.

• T. Gomez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), no. 1, 1--31. (math.AG/9911199)
• The appendix to A. Vistoli, Intersection theory on algebraic stacks, Invent. Math. 1997, contains a nice introduction.
• There are the first few chapters of a famous book project on stacks available on Andrew Kresch's homepage.
• The most complete reference is, of course: G. Laumon, L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3rd Series, 39, Springer-Verlag, Berlin, 2000, xii+208 pp. But this is usually considered as difficult to start with.
• For Vector bundles on algebraic curves, a lesser known reference are (incomplete) notes of a course by Faltings available at http://www.math.uni-bonn.de/people/fs/index/skripte.html.
• Finally there are some introductory notes on stacks available on my homepage (http://www.uni-due.de/~hm0002), as well as my (old) Diploma thesis on the stack of vector bundles on curves (in German), which might also serve as an introduction.