Abelian varieties and Prym varieties

Broadly speaking, I will be investigating the arithmetic and geometric applications of Abelian and Prym varieties. Initially I will be reading the paper by Beauville on the singularities of the Theta divisor, which provides a proof that some cubic hypersurface in P4 is not rational, although it is unirational. This was the first known counter-example to the Luroth problem. Prym varieties are featured in this paper, however the proof only works over fields of characteristic not equal to 2. We aim to extend the result and theory of Prym varieties used to this case.

Abelian varieties have been used with great success, in the context of Jacobian varieties, to prove results about rational points on curves of high genus (Falting's theorem). Prym varieties are a wider class of objects (than Jacobian varieties) whose arithmetic applications could tell us more about rational points.