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Deformations of isolated cyclic quotient singularities in arbitrary characteristic

Matthias Pfeifer

TL;DR

The paper addresses whether deformations of isolated cyclic quotient singularities preserve the same class across characteristics. It develops a framework of singularity-deformations and a normalization procedure via shifting automorphisms to analyze the generic fiber, ultimately showing that the geometric generic fiber contains at most isolated cyclic lrq singularities. This leads to a full proof of Riemenschneider's conjecture in arbitrary characteristic, extending classical results from equal-characteristic zero and providing a toric–lrq bridge to control singularities in deformations. The work also establishes semicontinuity of embedding dimensions and group-scheme lengths, with broad implications for the deformation theory of surface singularities. In short, the authors unify toric and lrq perspectives to resolve longstanding questions about deformations in positive and mixed characteristic.

Abstract

We show that toric surface singularities deform to toric surface singularities - both in equal and mixed characteristic. As an application, we establish Riemenschneiders conjecture that isolated cyclic quotient singularities of any dimension deform to isolated cyclic quotient singularities in equal and mixed characteristic.

Deformations of isolated cyclic quotient singularities in arbitrary characteristic

TL;DR

The paper addresses whether deformations of isolated cyclic quotient singularities preserve the same class across characteristics. It develops a framework of singularity-deformations and a normalization procedure via shifting automorphisms to analyze the generic fiber, ultimately showing that the geometric generic fiber contains at most isolated cyclic lrq singularities. This leads to a full proof of Riemenschneider's conjecture in arbitrary characteristic, extending classical results from equal-characteristic zero and providing a toric–lrq bridge to control singularities in deformations. The work also establishes semicontinuity of embedding dimensions and group-scheme lengths, with broad implications for the deformation theory of surface singularities. In short, the authors unify toric and lrq perspectives to resolve longstanding questions about deformations in positive and mixed characteristic.

Abstract

We show that toric surface singularities deform to toric surface singularities - both in equal and mixed characteristic. As an application, we establish Riemenschneiders conjecture that isolated cyclic quotient singularities of any dimension deform to isolated cyclic quotient singularities in equal and mixed characteristic.

Paper Structure

This paper contains 6 sections, 18 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Toric Surface Singularities
  3. Deformations
  4. Relation between proper and singularity-deformations
  5. Shifting
  6. Singularities of the Deformation

Key Result

Theorem 1.2

Let $X$ be an isolated cyclic lrq singularity and let $\mathcal{X}$ be a proper deformation of $X$ over a complete DVR $S$. Then the geometric generic fiber $\mathcal{X}_{\overline{\eta}}$ of $\mathcal{X}\to \text{Spec } S$ contains at worst isolated cyclic lrq singularities.

Theorems & Definitions (35)

  • Definition 1.1: LMM25
  • Theorem 1.2: Theorem \ref{['thm:cyc_to_cyc']}
  • Proposition 1.3: Proposition \ref{['prop:semicontinuity']}
  • Proposition 1.4: Proposition \ref{['prop:toric_eq_cyclic']}; LMM25
  • Proposition 2.1: LMM25
  • Theorem 2.2: Rie74
  • Theorem 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 25 more