Topics in Fano Varieties and Singularities

Abstract

In this thesis, we look at several problems in two areas of algebraic geometry: singularities and Fano varieties. From the singularities side, we examine the relationship between local fundamental groups and étale covers of the regular locus of a normal scheme. Here we are able to classify the obstructions to the map π_{1}(X_{reg}) \to π_{1}(X) being an isomorphism, and show that if they are finite there exists an étale cover of the regular locus of X where the maps are an isomorphism. In the area of Fano varieties, we study the notion of birational superrigidity. We show that under some extra conditions it implies K-stability a notion originating in the study of nice metrics on the Fano varieties. Moreover we show that hypersurfaces of sufficiently high dimension with respect to their index must satisfy some sort of rigidity assumption, restricting the base locus of any birational map to a Mori fiber space.