Jordan groups, conic bundles and abelian varieties

TL;DR

This paper investigates the Jordan property of birational automorphism groups of algebraic varieties, establishing conditions under which these groups are Jordan or not, especially for conic bundles over non-uniruled varieties and abelian surfaces.

Contribution

It proves that birational automorphism groups are Jordan for conic bundles over non-uniruled varieties, extending previous results and answering open questions for abelian surfaces.

Findings

Bir(X) is Jordan if X is a conic bundle over a non-uniruled variety.

Bir(X) is not Jordan if X is birational to a product of P^1 and an abelian variety.

Provides conditions distinguishing when Bir(X) has the Jordan property.

Abstract

A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $\mathcal{B}$ of $G$ contains a commutative subgroup $\mathcal{A}\subset \mathcal{B}$ such that $\mathcal{A}$ is normal in $\mathcal{B}$ and the index $[\mathcal{B}:\mathcal{A}] \le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the other hand, the second named author proved that $Bir(X)$ is not Jordan if $X$ is birational to a product of the projective line and a positive-dimensional abelian variety. We prove that $Bir(X)$ is Jordan if (uniruled) $X$ is a conic bundle over a non-uniruled variety $Y$ but is not birational to a product of $Y$ and the projective line. (Such a conic bundle exists only if $\dim(Y)\ge 2$.) When $Y$ is an abelian surface, this Jordaness property result gives an answer to a question of Prokhorov and Shramov.