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Equivariant geometry of singular cubic threefolds

Ivan Cheltsov, Yuri Tschinkel, Zhijia Zhang

TL;DR

This work investigates the linearizability of finite group actions on nodal cubic threefolds $X\subset \mathbb{P}^4$, focusing on the regime $2\le s(X)\le 9$ nodes and employing a toolkit that blends equivariant cohomology, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program. The authors classify linearizability case-by-case by node-count configurations (J1–J15) and invariant data $(s,d,p)$, using explicit geometric constructions (projections, unprojections, conic bundles) and degeneration techniques to derive obstructions or construct linearizations. Central contributions include detailed obstruction criteria (e.g., (H1)-cohomology, Burnside incompressibility, fixed-point criteria) and the identification of numerous nonlinearizable actions via intermediate Jacobian and Burnside analyses, complemented by birational rigidity results that certify when actions cannot be linearized. The results map out when $G$-actions on singular cubic threefolds are equivariantly birational to linear actions on projective space or to quadrics, with implications for equivariant rationality and Sarkisov theory in dimension three. Overall, the paper provides a nuanced, node-by-node landscape of linearizability and rigidity for finite group actions on nodal cubic threefolds, advancing the understanding of equivariant geometry in higher dimensions.

Abstract

We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.

Equivariant geometry of singular cubic threefolds

TL;DR

This work investigates the linearizability of finite group actions on nodal cubic threefolds , focusing on the regime nodes and employing a toolkit that blends equivariant cohomology, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program. The authors classify linearizability case-by-case by node-count configurations (J1–J15) and invariant data , using explicit geometric constructions (projections, unprojections, conic bundles) and degeneration techniques to derive obstructions or construct linearizations. Central contributions include detailed obstruction criteria (e.g., (H1)-cohomology, Burnside incompressibility, fixed-point criteria) and the identification of numerous nonlinearizable actions via intermediate Jacobian and Burnside analyses, complemented by birational rigidity results that certify when actions cannot be linearized. The results map out when -actions on singular cubic threefolds are equivariantly birational to linear actions on projective space or to quadrics, with implications for equivariant rationality and Sarkisov theory in dimension three. Overall, the paper provides a nuanced, node-by-node landscape of linearizability and rigidity for finite group actions on nodal cubic threefolds, advancing the understanding of equivariant geometry in higher dimensions.

Abstract

We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
Paper Structure (9 sections, 38 theorems, 222 equations)

This paper contains 9 sections, 38 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Obstructions to linearizability
  3. Two nodes
  4. Three nodes
  5. Four nodes
  6. Five nodes
  7. Six nodes
  8. Eight nodes
  9. Nine nodes

Key Result

Lemma 1.1

Let $X$ be a nodal cubic threefold. Let $G\subseteq \mathrm{Aut}(X)$ be such that it preserves a plane $\Pi\subset X$ and a line $l\subset X$, disjoint from $\Pi$. Then the $G$-action on $X$ is linearizable.

Theorems & Definitions (92)

  • Lemma 1.1
  • proof
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Example 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 82 more