Measuring rationality of Schwede--Takagi pairs

TL;DR

This paper characterizes rational singularities of pairs in the sense of Schwede--Takagi and introduces an invariant to measure their rationality failure.

Contribution

It extends existing characterizations of rational singularities to pairs and proposes a new categorical invariant for affine locally complete intersection pairs.

Findings

Provides a derived characterization of rational singularities for pairs.

Extends Lank--Venkatesh's characterization to pairs on normal varieties.

Introduces an invariant measuring the failure of rationality in specific cases.

Abstract

We begin by giving a derived characterization of rational singularities for pairs in the sense of Schwede--Takagi. This characterization extends a characterization of rational singularities due to Lank--Venkatesh to pairs on normal varieties over fields of characteristic zero. As an application, we introduce a categorical invariant that measures the failure of rationality for pairs on affine varieties that are locally complete intersections.