We prove that the group of birational transformations of a del Pezzo fibration of degree over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order . As a consequence we also obtain that the Cremona group of rank is not generated by birational maps preserving a rational fibration. Besides, the subgroup of 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 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 . Par conséquent, nous obtenons que le groupe de Cremona de rang n’est pas engendré par les applications birationnelles qui préservent une fibration rationnelles. De plus, le sous-groupe de engendré par tous les sous-groupes algébriques connexes est un sous-groupe distingué propre.
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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
@article{JEP_2020__7__1089_0, author = {J\'er\'emy Blanc and Egor Yasinsky}, title = {Quotients of groups of birational transformations of cubic del {Pezzo} fibrations}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1089--1112}, publisher = {\'Ecole polytechnique}, volume = {7}, year = {2020}, doi = {10.5802/jep.136}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.136/} }
TY - JOUR AU - Jérémy Blanc AU - Egor Yasinsky TI - Quotients of groups of birational transformations of cubic del Pezzo fibrations JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 1089 EP - 1112 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.136/ DO - 10.5802/jep.136 LA - en ID - JEP_2020__7__1089_0 ER -
%0 Journal Article %A Jérémy Blanc %A Egor Yasinsky %T Quotients of groups of birational transformations of cubic del Pezzo fibrations %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 1089-1112 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.136/ %R 10.5802/jep.136 %G en %F JEP_2020__7__1089_0
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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