Equivariant unirationality of Fano threefolds

TL;DR

The paper analyzes equivariant unirationality for finite group actions on Fano threefolds of index at least 2, introducing the central condition (A) that every abelian subgroup of G has a fixed point on X. It develops both general frameworks (torsors, twists, Amitsur/Bogomolov invariants) and constructive methods (invariant subvarieties, hyperplane sections, tangent-bundle techniques) to prove G-unirationality in many cases, and derives a stable linearization criterion for smooth quadric threefolds. It establishes G-unirationality for smooth and singular cubic threefolds under (A) with several explicit exceptions, and proves the equivalence for smooth intersections of two quadrics between G-unirationality and the fixed-point condition, using computational classifications of automorphism subgroups. The results advance the understanding of equivariant birational geometry in dimension three, link it to torsor twists and birational fixed-point criteria, and highlight remaining open cases (notably certain 3-groups on Fermat/Klein cubics and some actions on cubic threefolds).

Abstract

We study unirationality of actions of finite groups on Fano threefolds.

Theorems & Definitions (30)

• Theorem 2.1
• Lemma 2.2
• proof
• Example 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 3.1
• proof