Log canonical thresholds on Del Pezzo surfaces: Computation and applications

Log canonical thresholds are numerical invariants on Fano varieties introduced by Shokurov. Although they were originally introduced in the context of the Minimal Model Program they are not only important in birational geometry but also in complex differential geometry and stability theory. This invariant is hard to compute in general. Cheltsov, Park, Shramov and their students (among others) have computed log canonical thresholds for several classes of varieties such as (singular) del Pezzo surfaces, smooth Fano $3$-folds and certain examples in higher dimensions, always over algebraically closed fields in characteristic $0$.
In this talk I will introduce these ideas, reminding all basic concepts and explain why log canonical thresholds 'make sense' over algebraically closed fields of finite characteristic, as well as why we need to work 'harder'. This difference will be illustrated over non-singular algebraically closed Del Pezzo surfaces. Providing there is time left I will mention equivariant versions and computations over toric varieties.