Presentation of licentiate thesis in mathematics at Stockholm University

Respondent: Edoardo Mason
Title of the thesis: Stable sheaves and derived categories of surfaces with torsion canonical bundle
Supervisor: Sofia Tirabassi
Opponent: Gebhard Martin (Bonn)

Abstract:
The relation between moduli spaces of sheaves and the construction of equivalences between derived categories of algebraic varieties has proved to be extremely fruitful for solving problems regarding both topics. This work studies stable sheaves and derived categories in the case of surfaces with non-trivial torsion canonical bundle, namely Enriques and bielliptic surfaces.

In particular, we study the weak Brill-Noether property for moduli spaces of sheaves on complex bielliptic and Enriques surfaces, providing in the first case an application to birational geometry, namely the computation of the degree of irrationality of bielliptic surfaces. Our analysis uses both classical methods and new techniques arising from the theory of Bridgeland stability conditions on triangulated categories.

We also study twisted Fourier-Mukai partners of (quasi-)bielliptic surfaces over algebraically closed fields of critical characteristic. In particular, we determine the Brauer group of such surfaces and we compute the Hochschild cohomology of their twisted derived categories, allowing us to prove strong constraints on twisted derived equivalent (quasi-)bielliptic surfaces. Our strategy also involves the construction of relative Fourier-Mukai functors for surfaces along fibrations with fibers isomorphic to a cuspidal cubic curve, generalizing Bridgeland's argument for elliptic fibrations.