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Equivariant geometry of singular cubic threefolds, II

Ivan Cheltsov, Lisa Marquand, Yuri Tschinkel, Zhijia Zhang

TL;DR

This work completes a program to understand when finite group actions on singular cubic threefolds with isolated ADE singularities can be linearized or stabilized. By combining defect computations, automorphism classifications, and obstructions from Picard-group cohomology, equivariant specialization, and Burnside-type invariants, the authors map out a detailed landscape of linearizability across all singularity counts from two to six. The main contributions include a comprehensive defect table via projection, an explicit automorphism description for representative nonnodal cases (notably the $3\mathsf D_4$ example), and a case-by-case determination of (stable) linearizability driven by (H1), IJ, Sp, and Burnside tools. Collectively, the results advance the understanding of equivariant birational geometry for singular cubic threefolds and highlight the intricate interaction between singularities, symmetry, and birational properties.

Abstract

We study linearizability of actions of finite groups on cubic threefolds with nonnodal isolated singularities.

Equivariant geometry of singular cubic threefolds, II

TL;DR

This work completes a program to understand when finite group actions on singular cubic threefolds with isolated ADE singularities can be linearized or stabilized. By combining defect computations, automorphism classifications, and obstructions from Picard-group cohomology, equivariant specialization, and Burnside-type invariants, the authors map out a detailed landscape of linearizability across all singularity counts from two to six. The main contributions include a comprehensive defect table via projection, an explicit automorphism description for representative nonnodal cases (notably the example), and a case-by-case determination of (stable) linearizability driven by (H1), IJ, Sp, and Burnside tools. Collectively, the results advance the understanding of equivariant birational geometry for singular cubic threefolds and highlight the intricate interaction between singularities, symmetry, and birational properties.

Abstract

We study linearizability of actions of finite groups on cubic threefolds with nonnodal isolated singularities.
Paper Structure (9 sections, 52 theorems, 182 equations)

This paper contains 9 sections, 52 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. The Defect
  3. Automorphisms
  4. Picard groups and cohomology
  5. Two singular points
  6. Three singular points
  7. Four singular points
  8. Five Singular points
  9. Six singularities

Key Result

Theorem 2.1

Wall99 Consider a singular point $p\in C_q$, and assume $S_q$ is smooth.

Theorems & Definitions (106)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 96 more