The Tate-Shafarevich group of a polarised K3 surface
Algebraic Geometry
2025-07-31 v1
Abstract
In an earlier paper we generalised the notion of the Tate-Shafarevich group of an elliptic K3 surface to the Tate-Shafarevich group of a polarised K3 surface. In the present note, we complement the result by proving that the Tate-Shafarevich group of a polarised K3 surface (S,h) with h primitive parametrises bijectively all torsors for the Jacobian of the generic curve in the linear system |h| that admit a good hyperk\"ahler compactification. The result is seen as the analogue of the classical fact that the Tate-Shafarevich group of an elliptic K3 surface is the subgroup of the Weil-Ch\^atelet group of all twists that can be compactified to a K3 surface.
Keywords
Cite
@article{arxiv.2507.22703,
title = {The Tate-Shafarevich group of a polarised K3 surface},
author = {Daniel Huybrechts and Dominique Mattei},
journal= {arXiv preprint arXiv:2507.22703},
year = {2025}
}
Comments
19 pages. Comments are very welcome