Rational singularities and $q$-birational morphism

TL;DR

This paper generalizes rational singularities for reflexive sheaves, relates it to existing notions, introduces a dual concept $(B_{q+1})$, and proves results about $q$-birational morphisms.

Contribution

It extends the concept of rational singularities to reflexive sheaves and introduces a dual notion $(B_{q+1})$, linking it to $q$-birational morphisms.

Findings

Generalized rational singularities for reflexive sheaves.

Connected new notions with classical rational singularities.

Proved theorems on $q$-birational morphisms.

Abstract

In this paper, we generalize the notion of rational singularities for any reflexive sheaf of rank $1$, link our notion of rational singularities with the notion of rational singularities in [Kov11], and prove generalizations of standard facts about rational singularities. Moreover, by using a definition of non-rational locus, we introduce the notion of $(B_{q+1})$ as a dual notion of well-known Serre's notion of $(S_{q+1})$, and prove a theorem about $q$-birational morphisms.