Theta groups and products of abelian and rational varieties

TL;DR

This paper demonstrates that Jordan's theorem does not extend to the birational automorphism groups of certain algebraic varieties, specifically products of elliptic curves and the projective line, answering a question in algebraic geometry.

Contribution

It shows the failure of Jordan's theorem analogue for birational automorphism groups of specific product varieties, providing a negative result in algebraic geometry.

Findings

Jordan's theorem does not hold for these groups

Negative answer to Popov's question

Counterexamples in birational automorphism groups

Abstract

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question of V. L. Popov, arXiv:1001.1311 [math.AG].