Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inversion of adjunction for quotient singularities

Yusuke Nakamura, Kohsuke Shibata

Abstract

We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.

Inversion of adjunction for quotient singularities

Abstract

We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.

Paper Structure

This paper contains 19 sections, 40 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Log pairs
  5. Jet schemes and arc spaces for $k$-schemes
  6. Jet schemes and arc spaces for $k[t]$-schemes
  7. Codimension of cylinders in arc spaces
  8. Fundamental properties of the arc spaces of $k[t]$-schemes
  9. Dimension of the arc spaces of quotient varieties
  10. Denef and Loeser's theory for quotient varieties
  11. Lifting property of arcs on quotient varieties
  12. Arc spaces of quotient singularities
  13. Arc spaces of quotient varieties
  14. Arc space of hyperquotient singularities
  15. Arc spaces of quotient singularities
  16. ...and 4 more sections

Key Result

Theorem 1.2

Suppose a finite subgroup $G \subset {\rm GL}_N(k)$ acts on $\mathbb{A}_k^N$ freely in codimension one. Let $X := \mathbb{A}_k^N / G$ be the quotient variety. Let $Y$ be a subvariety of $X$ of codimension $c$ which has only klt singularities, and let $\mathfrak{a}$ be an $\mathbb{R}$-ideal sheaf on is lower semi-continuous, where we denote by $|Y|$ the set of all closed points of $Y$ with the Zar

Theorems & Definitions (109)

  • Conjecture 1.1: LSC conjecture
  • Theorem 1.2: $=$ Theorem \ref{['thm:LSC_general']}
  • Conjecture 1.3: PIA conjecture, 92
  • Theorem 1.4: $=$ Corollary \ref{['cor:PIA_general']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • ...and 99 more