  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.
