On the Geometry of Principal Homogeneous Spaces
A. J. de Jong, Robert Friedman

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 be a curve defined over an algebraically closed field and let be an elliptic surface with base curve . We investigate the geometry of everywhere locally trivial principal homogeneous spaces for , i.e. elements of the Tate-Shafarevich group. If is such a principal homogeneous space of order , we find strong restrictions on the bundle over into which embeds. Examples for small values of show that, in at least some cases, these restrictions are sharp. Finally, we determine these bundles in case has characteristic zero, , and is generic in a suitable sense.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Finite Group Theory Research
