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On linear $α_p$-quotients

Quentin Posva, Takehiko Yasuda

Abstract

We study ``linear" $α_p$-actions on affine spaces and the associated quotient singularities, using explicit stacky resolutions. We describe when the quotient singularities are log canonical, canonical or terminal, and we compute their stringy motivic invariants. The second author and Fabio Tonini conjectured that these invariants coincide with those of linear $\bZ/p$-quotients: our approach reduces this conjecture to an equality of explicit multi-sets, which we check for a large number of primes using a computer software.

On linear $α_p$-quotients

Abstract

We study ``linear" -actions on affine spaces and the associated quotient singularities, using explicit stacky resolutions. We describe when the quotient singularities are log canonical, canonical or terminal, and we compute their stringy motivic invariants. The second author and Fabio Tonini conjectured that these invariants coincide with those of linear -quotients: our approach reduces this conjecture to an equality of explicit multi-sets, which we check for a large number of primes using a computer software.
Paper Structure (19 sections, 38 theorems, 125 equations)

This paper contains 19 sections, 38 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Notations and conventions
  5. Deligne--Mumford stacks
  6. 1-foliations
  7. Stringy motivic invariant
  8. Degeneration of linear $\mathbb{Z}/p$-actions into $\alpha_p$-actions
  9. Partial resolutions of linear $\alpha_p$-quotients
  10. Notations
  11. Weighted blow-up of the ramification locus
  12. Irreducible case
  13. General case
  14. MMP singularities of linear $\alpha_p$-quotients
  15. Stratification of the partial resolution
  16. ...and 4 more sections

Key Result

Theorem 1.0.1

Let $\mathbb{D}_\mathbf{d}=\sum_\lambda \frac{d_\lambda(d_\lambda+1)}{2}$. Then $\mathbb{A}/(\alpha_p,\mathbf{d})$ is lc if and only if $\mathbb{D}_\mathbf{d}\geq p-1$, is canonical if and only if $\mathbb{D}_\mathbf{d}\geq p$, and is terminal if and only if $\mathbb{D}_\mathbf{d}\geq p+1$.

Theorems & Definitions (77)

  • Theorem 1.0.1: \ref{['thm:MMP_sing_alpha_p_qt']}
  • Theorem 1.0.2: \ref{['thm:mst_alpha_p_qt']}
  • Conjecture 1: Tonini_Yasuda_Motivic_McKay_cor_for_alpha_p
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 67 more