Rational singularities associated to pairs

TL;DR

This paper extends the concept of rational singularities to pairs involving a variety, an ideal, and a real number, establishing foundational results and analogs to log terminal pairs, including in positive characteristic.

Contribution

Introduces a new notion of rational singularities for pairs and extends key properties and results from classical rational singularities to this broader context.

Findings

Most standard rational singularity results extend to pairs.

Analogues of log terminal pair results are established.

A positive characteristic analogue is defined and explored.

Abstract

In this paper we introduce a notion of rational singularities associated to pairs $(X, \ba^t)$ where $X$ is a variety, $\ba$ is an ideal sheaf and $t$ is a nonnegative real number. We prove that most standard results about rational singularities extend to this context. We also show that some results commonly associated with log terminal pairs have analogs in this context, including results related to inversion of adjunction. A positive characteristic analogue of rational singularities of pairs is also defined and explored.