Abelian varieties and Prym varieties

概括地说，我将研究Abelian和Prym簇的算术和几何应用。首先，我将阅读Beauville关于Theta因子奇点的论文，该论文证明了P4中的一些三次超曲面不是有理的，尽管它是单调的。这是已知的第一个针对Luroth问题的反例。Prym变种是本文的特征，但证明只适用于特征不等于2的域。我们的目的是将Prym变种的结果和理论推广到这种情况。Abel变元在Jacobian变元的背景下，被成功地用来证明关于高亏格曲线上有理点的结果(Falting定理)。Prym变数是一类更广泛的对象(比Jacobian变数)，其算术应用程序可以告诉我们更多关于有理点的信息。

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.