The special Brauer group and twisted Picard varieties

TL;DR

This paper extends the concept of the Tate-Shafarevich group to K3 surfaces with linear systems using Grothendieck's special Brauer group, facilitating the study of moduli spaces of twisted sheaves.

Contribution

It introduces a new framework for understanding the Tate-Shafarevich group of K3 surfaces with linear systems via the special Brauer group, generalizing previous notions.

Findings

Provides an efficient method for handling moduli spaces of twisted sheaves.

Establishes a generalized Tate-Shafarevich group for K3 surfaces with linear systems.

Connects the special Brauer group to the geometry of twisted sheaves.

Abstract

We generalise the notion of the Tate-Shafarevich group of an elliptic K3 surface with a section to the Tate-Shafarevich group of a K3 surface endowed with a linear system. The construction, which uses Grothendieck's special Brauer group, provides an efficient way to deal with moduli spaces of twisted sheaves supported on curves in a K3 surface.