Del Pezzo surfaces with Du Val singularities

Abstract

A lot of attention has been drawn recently to global log canonical thresholds of Fano varieties, which are algebraic counterparts of the α-invariant of Tian for smooth Fano varieties. In particular, global log canonical thresholds are related to the existence of Kahler-Einstein metrics on Fano varieties. The purpose of this thesis is to apply techniques from singularity theory in order to compute the global log canonical thresholds of all Del Pezzo surfaces of degree 1 with Du Val singularities, as well as the global log canonical thresholds of all Del Pezzo surfaces of Picard rank 1 with Du Val singularities. As a consequence, it is proven that Del Pezzo surfaces of degree 1 with Du Val singularities admit a Kahler-Einstein metric if the singular locus consists of only A1, or A3, or A4 type Du Val singular points.