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Equivariant geometry of cubic threefolds with non-isolated singularities

Ivan Cheltsov, Lisa Marquand, Yuri Tschinkel, Zhijia Zhang

TL;DR

The paper extends the study of linearizability of finite group actions from cubic threefolds with isolated singularities to those with non-isolated singularities, classifying the geometric types of singularities into line, conic, plane, and chordal. It develops and applies birational-tools to reduce linearization questions to base actions and explicit birational models, proving that actions on line, chordal, and plane singularities are linearizable, while most conic cases are not, though all such threefolds are $G$-unirational. The chordal cubic case is shown to have linearizable finite group actions despite the full automorphism group not acting linearly, due to a PGL2-equivariant fibration to a Veronese base. Across the various geometries, unprojection and explicit normal forms play central roles in achieving linearizability results, while unirationality persists for all finite group actions. The results illuminate the interplay between equivariant geometry and arithmetic considerations over nonclosed fields, and provide techniques applicable to moduli and birational classification of singular cubic varieties.

Abstract

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

Equivariant geometry of cubic threefolds with non-isolated singularities

TL;DR

The paper extends the study of linearizability of finite group actions from cubic threefolds with isolated singularities to those with non-isolated singularities, classifying the geometric types of singularities into line, conic, plane, and chordal. It develops and applies birational-tools to reduce linearization questions to base actions and explicit birational models, proving that actions on line, chordal, and plane singularities are linearizable, while most conic cases are not, though all such threefolds are -unirational. The chordal cubic case is shown to have linearizable finite group actions despite the full automorphism group not acting linearly, due to a PGL2-equivariant fibration to a Veronese base. Across the various geometries, unprojection and explicit normal forms play central roles in achieving linearizability results, while unirationality persists for all finite group actions. The results illuminate the interplay between equivariant geometry and arithmetic considerations over nonclosed fields, and provide techniques applicable to moduli and birational classification of singular cubic varieties.

Abstract

We study linearizability of actions of finite groups on cubic threefolds with non-isolated singularities.
Paper Structure (6 sections, 12 theorems, 58 equations)

This paper contains 6 sections, 12 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Tools from birational geometry
  3. Line
  4. Conic
  5. The chordal cubic
  6. Plane

Key Result

Proposition 2.1

Let $X={\mathbb P}_{B}({\mathcal{E}})$ be the projectivization of a vector bundle ${\mathcal{E}}$ of rank $n+1$ over a smooth projective irreducible variety $B$, and $\pi:X\to B$ the associated ${\mathbb P}^n$-bundle. Assume that $X$ carries a regular action of a finite group $G$ such that ${\mathca

Theorems & Definitions (26)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 16 more