The Tate-Shafarevich group of a polarised K3 surface

TL;DR

This paper extends the concept of the Tate-Shafarevich group to polarized K3 surfaces, establishing a bijective correspondence with certain torsors that admit hyperkähler compactifications, analogous to classical elliptic surface results.

Contribution

It proves that the Tate-Shafarevich group of a polarized K3 surface parametrizes torsors with hyperkähler compactifications, generalizing previous notions and drawing parallels with elliptic K3 surfaces.

Findings

Tate-Shafarevich group bijects with torsors admitting hyperkähler compactifications

Generalization of Tate-Shafarevich group concept to polarized K3 surfaces

Analogy with classical elliptic K3 surface results

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.