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arXiv · 2507.22703

The Tate-Shafarevich group of a polarised K3 surface

Daniel Huybrechts·Dominique Mattei
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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.

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Daniel Huybrechts, Dominique Mattei. 2025-07-30. The Tate-Shafarevich group of a polarised K3 surface. https://arxiv.org/abs/2507.22703

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