Relations in the Cremona group over perfect fields

TL;DR

This paper constructs new normal subgroups of the Cremona group over certain perfect fields and provides an elementary proof that this group is not simple, extending recent results to new field conditions.

Contribution

It introduces new normal subgroups for the Cremona group over perfect fields with specific properties and offers an elementary proof of its non-simplicity, broadening understanding of its algebraic structure.

Findings

Constructed new normal subgroups of the Cremona group over perfect fields.

Provided an elementary proof of the group's non-simplicity.

Extended non-simplicity results to fields with certain algebraic properties.

Abstract

For perfect fields $k$ satisfying $[\bar k:k]>2$, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank $n$ over (subfields of) the complex numbers is not simple for $n\geq3$.