Algebra seminar

Julian Rosen, Department of Pure Mathematics, University of Waterloo

"Curves on abelian surfaces"

This talk will begin with an introduction to abelian varieties, which are projective varieties having the structure of an abelian group. Abelian varieties of dimension 1 are elliptic curves. Our focus will be on abelian varieties of dimension 2.

We will discuss a correspondence between certain classes of line bundles on an abelian surface and integral quadratic forms. This correspondence can be used to determine which genera of curves can be embedded in various abelian surfaces, and which abelian surfaces embed in the projective space P^4 (every abelian surface embeds in P^5). Some of the answers are quite surprising.

This talk is based on joint work with Ari Shnidman.