On the geometry of principal homogeneous spaces

Abstract

Let $B$ be a curve defined over an algebraically closed field $k$ and let $\pi\colon \ X\to B$ 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 of the generic fiber of $\pi$. If $Y$ is such a principal homogeneous space of order~$n$, we find strong restrictions on the ${\Bbb P}^{n-1}$ bundle over $B$ into which $Y$ embeds. Examples for small values of $n$ show that, at least in some cases, these restrictions are sharp. Finally, we determine these bundles in case $k$ has characteristic zero, $B = {\Bbb P}^1$, and $X$ is generic in a suitable sense.