On the Geometry of Principal Homogeneous Spaces

TL;DR

This paper studies the geometry of principal homogeneous spaces related to elliptic surfaces over curves, revealing restrictions on their embeddings and characterizing these structures in specific cases, especially over complex projective lines.

Contribution

It provides new restrictions on embeddings of principal homogeneous spaces into projective bundles and characterizes these bundles for generic elliptic surfaces over the projective line in characteristic zero.

Findings

Restrictions on $ ext{P}^{n-1}$ bundles for principal homogeneous spaces

Examples demonstrating sharpness of restrictions for small $n$

Explicit determination of bundles over $ ext{P}^1$ in characteristic zero

Abstract

Let $B$ be a curve defined over an algebraically closed field $k$ and let $X\to B$ be an elliptic surface with base curve $B$. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for $X$, i.e. elements of the Tate-Shafarevich group. If $Y$ is such a principal homogeneous space of order $n$, we find strong restrictions on the $\mathbb{P}^{n-1}$ bundle over $B$ into which $Y$ embeds. Examples for small values of $n$ show that, in at least some cases, these restrictions are sharp. Finally, we determine these bundles in case $k$ has characteristic zero, $B = \mathbb{P}^1$, and $X$ is generic in a suitable sense.