Brauer Groups of Bielliptic Surfaces and Twisted Derived Equivalences
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This thesis focuses on two interrelated projects. The first project concerns the study of bielliptic surfaces, their Brauer groups and the pullback maps from their Brauer groups to those of their canonical covers. We prove results classifying injectivity and triviality of these maps. In order to do this, we provide some results of a very classical flavor: first we give generators for the torsion of the second integral cohomology of bielliptic surfaces, and secondly we give structure theorems for the Picard group of the product of two elliptic curves. The second project revolves around the study of the twisted derived category of bielliptic surfaces. We expose some of the structure of these derived categories, and prove that an untwisted bielliptic surface does not admit any twisted Fourier-Mukai partner. This is done utilizing the results of the first part and the geometry of moduli spaces of sheaves.