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Rational singularities and $q$-birational morphism

Donghyeon Kim

TL;DR

The paper generalizes rational singularities to reflexive sheaves of rank $1$ on normal varieties, linking the new framework to Kovrational while working for any algebraically closed field and $q\ge 2$. It develops duality-based notions $(KV_q)$, $(RS_q)$, $(S_{q+1})$, and $(B_{q+1})$, establishing equivalences that connect the dual $\mathcal{F}^D$ to the original $\mathcal{F}$, and clarifies when rational singularities hold in this broader setting. A key contribution is introducing $(B_{q+1})$ as a dual analog to $(S_{q+1})$, with a main theorem showing that, under $(RS_q)$, $\mathcal{F}^D$ being $(B_{q+1})$ is equivalent to $\mathcal{F}$ being $(S_{q+1})$. The results yield CM-criteria for normal $\mathbb{Q}$-factorial varieties under $(R_q)$ and divisors $(KV_q)$, along with vanishing theorems for $q$-birational morphisms, providing a robust toolkit for birational geometry in the singular setting.

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.

Rational singularities and $q$-birational morphism

TL;DR

The paper generalizes rational singularities to reflexive sheaves of rank on normal varieties, linking the new framework to Kovrational while working for any algebraically closed field and . It develops duality-based notions , , , and , establishing equivalences that connect the dual to the original , and clarifies when rational singularities hold in this broader setting. A key contribution is introducing as a dual analog to , with a main theorem showing that, under , being is equivalent to being . The results yield CM-criteria for normal -factorial varieties under and divisors , along with vanishing theorems for -birational morphisms, providing a robust toolkit for birational geometry in the singular setting.

Abstract

In this paper, we generalize the notion of rational singularities for any reflexive sheaf of rank , 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 as a dual notion of well-known Serre's notion of , and prove a theorem about -birational morphisms.
Paper Structure (6 sections, 26 theorems, 69 equations)

This paper contains 6 sections, 26 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reflexive sheaves
  4. Rational singularities
  5. Notion of $(B_{q+1})$
  6. $q$-birational morphisms

Key Result

Theorem 1.5

Let $X$ be any normal variety over a characteristic $0$ field, $\Delta$ any effective $\mathbb{Q}$-Weil divisor such that $(X,\Delta)$ is dlt and $D$ any $\mathbb{Q}$-Cartier Weil divisor on $X$. Then $D$ has rational singularities.

Theorems & Definitions (71)

  • Definition 1.1: See huybrechts2010geometry, Definition 1.1.7
  • Definition 1.3
  • Definition 1.4: See Definition 2.5, and Theorem 2.9 in Kovrational
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • Remark 1.10
  • Lemma 1.11
  • ...and 61 more