On the Geometry of Principal Homogeneous Spaces

2008/10/15 by A. J. de Jong, Robert Friedman, de Jong, A. J. +1
Mathematics · #14J27 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.0810.2687

openalex publication_date 2008/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B be a curve defined over an algebraically closed field k and let X→ 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 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 = ℙ¹, and X is generic in a suitable sense.

Citations

• Vector Bundles Over an Elliptic Curve

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