Results on singularities of pairs

Abstract

Singularities of algebraic varieties have been studied extensively, and recently also the properties of singularities of pairs have been investigated. This thesis presents several results on singularities of different kinds of pairs. The simplest pairs are of the form (P^2,C) where C is a plane curve: the first chapter presents an approach to classifying isomorphism classes for ordinary multiple points of plane curves. The next chapter is an introduction to singularities in algebraic geometry, including the fundamental definitions and properties of rational pairs, introduced by Kollár and Kovács in 2013. The final three chapters contain novel results on log canonical centers, rationalizing divisors and images of rational pairs under certain morphisms.