The special Brauer group and twisted Picard varieties

Abstract

We generalise the notion of the Tate–Šafarevič group $Ш(S/{\mathbb{P}}^1)$ of an elliptic K3 surface $S\to {\mathbb{P}}^1$ with a section to $Ш(S,h)$ of a K3 surface $S$ endowed with a linear system $|h|$. 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 $|h|$.