Derived categories of twisted sheaves on elliptic threefolds (Q2781433)

This paper gives an equivalence between the derived category of sheaves on an elliptic threefold \(X \to S\) without a section (here called a ``generic elliptic threefold'') and a derived category of twisted sheaves on any analytic small resolution \(\overline{J}\) of its relative Jacobian \(J \to S\). The twist depends on an element of the Brauer group of \(\overline{J}\) which is the obstruction to gluing universal sheaves on patches of \(X\) arising from open subsets of \(\overline{J}\). The replacement of \(J\) by the analytic resolution and the twisting together give a substitute construction \(\mathcal{U}\) for a universal sheaf on \(X \times_{S} J\), which does not exist in the situation considered by the author. The equivalence of derived categories is achieved by extending \(\mathcal{U}\) by \(0\) along a certain inclusion.