Mathematics

Picking up the thread from last week, we will give another (more numerical) characterisation of rational surface singularities. We will then see how this, combined with the Artin Contraction Theorem (a slight generalisation of the classical Castelnuovo Contraction Theorem) can often reduce the study of rational surface singularities to simply keeping track of the combinatorics of the exceptional locus of a resolution. We will exhibit this phenomenon in the elegant case of Du Val singularities. If time permits, we will sketch the beautiful connection of Du Val singularities with regular solids in three dimensions, including the Platonic solids that have been studied since Classical Antiquity.