Quotients of groups of birational transformations of cubic del Pezzo fibrations

Blanc, Jérémy; Yasinsky, Egor

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1089-1112.

Nous démontrons que le groupe des transformations birationnelles d’une fibration de del Pezzo de degré $3$ sur une courbe n’est pas simple, en donnant un homomorphisme de groupes surjectif vers un produit libre d’une infinité de groupes d’ordre $2$. Par conséquent, nous obtenons que le groupe de Cremona de rang $3$ n’est pas engendré par les applications birationnelles qui préservent une fibration rationnelles. De plus, le sous-groupe de $\mathrm{Bir}\left({\mathbb{P}}^3\right)$ engendré par tous les sous-groupes algébriques connexes est un sous-groupe distingué propre.

We prove that the group of birational transformations of a del Pezzo fibration of degree $3$ over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order $2$. As a consequence we also obtain that the Cremona group of rank $3$ is not generated by birational maps preserving a rational fibration. Besides, the subgroup of $\mathrm{Bir}\left({\mathbb{P}}^3\right)$ generated by all connected algebraic subgroups is a proper normal subgroup.

Classification : 14E07, 14E05, 14E30, 14J45, 14M22
Keywords: Del Pezzo fibrations, Cremona group, group homomorphisms, group quotients, birational transformations, genus
Mot clés : Fibrations de del Pezzo, groupe de Cremona, homomorphismes de groupes, quotients de groupes, transformations birationnelles, genre

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