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
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.
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.
- 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
- 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
- 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.