On G-birational rigidity of projective spaces

TL;DR

This paper investigates the finite subgroups of automorphisms of projective spaces that preserve birational rigidity, establishing finiteness results for such groups and identifying specific superrigid cases.

Contribution

It proves finiteness of conjugacy classes of G-subgroups for projective spaces of dimension at least 3 and identifies a unique superrigid subgroup in dimension 4.

Findings

Finite subgroups of Aut(ℙ^n) preserving G-birational rigidity are finite in number for n≥3.

ℙ^4 is G-birationally superrigid when G is isomorphic to PSp_4(𝔽_3).

The automorphism groups contain only finitely many conjugacy classes of such subgroups.

Abstract

In this paper, we study finite subgroups $G\subset\mathrm{Aut}(\mathbb{P}^n)$ such that $\mathbb{P}^n$ is $G$-birationally rigid. For each $n\geqslant 3$, we prove that $\mathrm{Aut}(\mathbb{P}^n)$ contains at most finitely many such subgroups up to conjugation. For $n=4$, we prove that $\mathbb{P}^4$ is $G$-birationally superrigid if $G\simeq\mathrm{PSp}_{4}(\mathbf{F}_3)$.