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Quotient singularities by permutation actions are canonical

Takehiko Yasuda

TL;DR

The work addresses when quotient varieties arising from permutation representations are canonical and how the associated log pairs behave in various characteristics. It leverages the wild McKay correspondence and motivic integration to translate torsor data over the punctured formal disk into global singularity statements, producing canonical singularities in arbitrary characteristic and LC/KLT criteria depending on the characteristic. Key contributions include precise Gorenstein-index computations, dimension bounds for loci in P-moduli spaces, and refined estimates in low characteristics when transpositions are absent. These results extend known positive-characteristic phenomena (like F-purity) to canonical and log-terminal contexts for permutation actions, with implications for possible extensions to broader monomial representations.

Abstract

The quotient variety associated to a permutation representation of a finite group has only canonical singularities in arbitrary characteristic. Moreover, the log pair associated to such a representation is Kawamata log terminal except in characteristic two, and log canonical in arbitrary characteristic.

Quotient singularities by permutation actions are canonical

TL;DR

The work addresses when quotient varieties arising from permutation representations are canonical and how the associated log pairs behave in various characteristics. It leverages the wild McKay correspondence and motivic integration to translate torsor data over the punctured formal disk into global singularity statements, producing canonical singularities in arbitrary characteristic and LC/KLT criteria depending on the characteristic. Key contributions include precise Gorenstein-index computations, dimension bounds for loci in P-moduli spaces, and refined estimates in low characteristics when transpositions are absent. These results extend known positive-characteristic phenomena (like F-purity) to canonical and log-terminal contexts for permutation actions, with implications for possible extensions to broader monomial representations.

Abstract

The quotient variety associated to a permutation representation of a finite group has only canonical singularities in arbitrary characteristic. Moreover, the log pair associated to such a representation is Kawamata log terminal except in characteristic two, and log canonical in arbitrary characteristic.
Paper Structure (13 sections, 15 theorems, 65 equations)

This paper contains 13 sections, 15 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singularities in the minimal model program
  4. Stringy motives
  5. P-Moduli spaces of formal torsors
  6. The wild McKay correspondence
  7. Permutation actions and dimension estimates
  8. The Gorenstein index of $\boldsymbol{X}$
  9. Dimensions of loci in moduli spaces
  10. Better estimations derived from the non-existence of transposition
  11. Proofs of Theorems \ref{['thm:main-non-log']} and \ref{['thm:main-log']}
  12. Proof of Theorem \ref{['thm:main-log']}
  13. Proof of Theorem \ref{['thm:main-non-log']}

Key Result

Theorem 1.2

Suppose that $G$ acts on $V$ by permutations of coordinates. Then, the quotient variety $X$ has only canonical singularities.

Theorems & Definitions (29)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: cf. yasuda2024motivic
  • Proposition 2.4: cf. yasuda2024motivic
  • proof
  • Proposition 2.5
  • proof
  • ...and 19 more