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On the geometry of principal homogeneous spaces
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 3, June 2011
- pp. 753-796
- 10.1353/ajm.2011.0020
- Article
- Additional Information
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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.