Some Abstracts

9/21 This lecture will cover basic properties of vector bundles including stability. I will present vector bundles as extensions of smaller rank bundles and show applications.

10/12 We will discuss the concept of quotient (orbit space, categorical, geometric and good quotients) and its use in the construction of moduli spaces. We will describe in particular the construction of affine and projective quotients using geometric invariant theory. The content will be similar to that of Lecture 1 of my "Polish" notes (see http://www.mimuw.edu.pl/~jarekw/EAGER/Lukecin02).

2/23 Title: Dolgachev's conjecture on the moduli space of rank 3 bundles on a genus 2 curve. Abstract: For a curve of genus 2, the moduli space of rank 3 vector bundles with trivial determinant can be realized as a double cover of the 8-dimensional projective space branched along a sextic hypersurface. On the other hand, Coble proved a century ago that there exists a unique cubic hypersurface in 8-space which is (1) singular along the Jacobian variety of the curve (which is embedded using the system of 3-theta functions) and (2) invariant under the natural action coming from torsion points of order 3 on the Jacobian. Dolgachev has made the striking conjecture than these two seemingly completely unrelated hypersurfaces are in fact projectively dual to each other. I will discuss the background of the problem and present a proof of Dolgachev's conjecture.