On rationalizing divisors

TL;DR

This paper investigates conditions under which a normal variety admits a rationalizing divisor, extending the concept of rational singularities to pairs and providing criteria for cones.

Contribution

It introduces a criterion for cones to possess rationalizing divisors and links their existence to the rational singularity locus of the variety.

Findings

Criteria for cones to have rationalizing divisors

Relation between rationalizing divisors and rational singularity locus

Extension of rational singularities to pairs

Abstract

Rational pairs generalize the notion of rational singularities to reduced pairs $(X,D)$. In this paper we deal with the problem of determining whether a normal variety $X$ has a rationalizing divisor, i.e. a reduced divisor $D$ such that $(X, D)$ is a rational pair. We give a criterion for cones to have a rationalizing divisor, and relate the existence of such a divisor to the locus of rational singularities of a variety.