On Fano varieties of low Picard number with torus action

Abstract

This thesis contributes to the explicit classification of Fano varieties of Picard number one and two. In the first chapter we give sharp upper bounds on various geometric invariants of fake weighted projective spaces, only depending on the dimension and the Gorenstein index. Moreover, we present an efficient procedure for explicitly classifying fake weighted projective spaces of any dimension and Gorenstein index and we carry out the classification up to dimension four for various Gorenstein indices. Chapter two is devoted to the classification of the non-toric, Q-factorial, log terminal, Gorenstein Fano threefolds of Picard number one that admit an effective action of a two-dimensional torus. Finally, in chapter three we classify the locally factorial Fano fourfolds of Picard number two with a hypersurface Cox ring that admit an effective action of a three-dimensional torus.