Deformation invariance of rational pairs

Abstract

Rational pairs, recently introduced by Kollár and Kovács, generalize rational singularities to pairs (X,D). Here X is a normal variety and D is a reduced divisor on X. Integral to the definition of a rational pair is the notion of a thrifty resolution, also defined by Kollár and Kovács, and in order to work with rational pairs it is often necessary to know whether a given resolution is thrifty. In this dissertation I present several foundational results that are helpful for identifying thrifty resolutions and analyzing their behavior. In 1978, Elkik proved that rational singularities are deformation invariant. The main result of this dissertation is an analogue of this theorem for rational pairs: given a flat family X over S and a Cartier divisor D on X, if the fibers over a smooth point s in S form a rational pair, then (X,D) is also rational near the fiber Xs.