Relations in the Cremona group over a perfect field

Abstract

For perfect fields k satisfying [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 3.Résumé. -Pour les corps parfaits k qui satisfont [k : k] > 2, nous construisons de nouveaux sous-groupes distingués du groupe de Cremona du plan et nous donnons une preuve élémentaire de sa non-simplicité en suivant la mélodie de la preuve récente de Blanc, Lamy et Zimmermann du fait que le groupe de Cremona de rang n sur les (sous-corps des) nombres complexes n'est pas simple pour n 3.

Rigid birational involutions of ℙ 3 and cubic threefolds

“…The Cremona group in dimension 2 over any field k is known to be non-simple (see [CL13,Lon16]) i.e., it admits non-trivial homomorphisms to other groups. Recently, many families of such homomorphisms were constructed: for example in dimension 2 over a perfect field by [LZ20,Sch22] and over a subfield of the complex numbers by [BY20] in dimension 3 and by [BLZ21] in dimension greater or equal to 3. Among other important consequences, the examples in the latter case proved for the first time the non-simplicity of the Cremona group in dimension greater or equal to 3.…”