Quotients of groups of birational transformations of cubic del Pezzo fibrations
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 1089-1112.

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 Bir( 3 ) generated by all connected algebraic subgroups is a proper normal subgroup.

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 Bir( 3 ) engendré par tous les sous-groupes algébriques connexes est un sous-groupe distingué propre.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.136
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

Jérémy Blanc 1; Egor Yasinsky 1

1 Universität Basel, Departement Mathematik und Informatik Spiegelgasse 1, CH-4051 Basel, Switzerland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Jérémy Blanc; Egor Yasinsky. Quotients of groups of birational transformations of cubic del Pezzo fibrations. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 1089-1112. doi : 10.5802/jep.136. https://jep.centre-mersenne.org/articles/10.5802/jep.136/

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