On the Geometry of Principal Homogeneous Spaces

Details On the Geometry of Principal Homogeneous Spaces On the Geometry of Principal Homogeneous Spaces

2008-10-15

arxiv.org/abs/0810.2687v1

Mathematics

Algebraic Geometry

49 pages

Scientific paper

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.

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