# Jordan properties of automorphism groups of certain open algebraic varieties

## TL;DR

This paper proves that the automorphism groups of certain open algebraic varieties, specifically those birational to a product of a smooth projective variety without rational curves and a projective line, are Jordan groups.

## Contribution

It establishes the Jordan property for automorphism groups of a new class of open algebraic varieties under specific geometric conditions.

## Key findings

- Automorphism groups are Jordan for varieties birational to A×P^1 with A containing no rational curves.
- Existence of a uniform bound J for the index of abelian subgroups in finite automorphism subgroups.
- Extends Jordan property results to a broader class of open algebraic varieties.

## Abstract

Let $W$ be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that $W$ is birational to a product of a smooth projective variety $A$ and the projective line. We prove that if $A$ contains no rational curves then the automorphism group $G:=Aut(W)$ of $W$ is Jordan. That means that there is a positive integer $J=J(W)$ such that every finite subgroup $\mathcal{B}$ of ${G}$ contains a commutative subgroup $\mathcal{A}$ such that $\mathcal{A}$ is normal in $\mathcal{B}$ and the index $[\mathcal{B}:\mathcal{A}] \le J$ .