Quotients of groups of birational transformations of cubic del Pezzo fibrations

Blanc, Jérémy; Yasinsky, Egor

doi:10.5802/jep.136

Cited by 9 publications

(6 citation statements)

References 26 publications

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“…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.…”

Section: Homomorphisms From the Cremona Group And Free Product Structurementioning

confidence: 99%

Rigid birational involutions of ℙ 3 and cubic threefolds

Zikas¹

2023

Journal De L’École Polytechnique — Mathématiques

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We construct families of birational involutions on P 3 or on a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.Résumé (Involutions birationnelles rigides de P 3 et de cubiques lisses de dimension 3)Nous construisons des familles d'involutions birationnelles sur P 3 ou sur une cubique lisse de dimension 3 qui ne s'intègrent pas dans une relation élémentaire non triviale de liens de Sarkisov. En conséquence, nous construisons de nouveaux homomorphismes à partir de leur groupe de transformations birationnelles, redémontrant de manière effective leur non-simplicité. Nous prouvons également que ces groupes admettent une structure de produit libre. Enfin, nous produisons des automorphismes de ces groupes qui ne sont pas engendrés par des automorphismes intérieurs et des automorphismes de corps.

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Section: Homomorphisms From the Cremona Group And Free Product Structurementioning

confidence: 99%

Rigid birational involutions of ℙ 3 and cubic threefolds

Zikas¹

2023

Journal De L’École Polytechnique — Mathématiques

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“…While Bir(ℙ 2 𝐤 ) is compactly presented in the Euclidean topology by a quadratic involution of ℙ 2 𝐤 and a compact subset of Aut(ℙ 2 ) [18, Theorem A], the group Bir(ℙ 𝑛 𝐤 ), 𝑛 ⩾ 3 is not generated by a compact subset (Proposition 2.6 (7)). In fact, Bir(ℙ 3 ℂ ) is not even generated by its algebraic subgroups [2,Theorem C]. The Euclidean topology on Cremona groups is largely unstudied and results can be found in [1,3,16,18].…”

Section: Introductionmentioning

confidence: 99%

Properties of the Cremona group endowed with the Euclidean topology

Bergner¹,

Zimmermann

2023

Bulletin of London Math Soc

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Consider a Cremona group endowed with the Euclidean topology introduced by Blanc and Furter. It makes it a Hausdorff topological group that is not locally compact nor metrisable. We show that any sequence of elements of the Cremona group of bounded order that converges to the identity is constant. We use this result to show that Cremona groups do not contain any non-stationary sequence of subgroups converging to the identity. We also show that, in general, paths in a Cremona group do not lift and do not satisfy a property similar to the definition of morphisms to a Cremona group.

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“…Observing similarities between the plane Cremona group over non-closed fields and the Cremona groups in high dimensions, Blanc, Lamy and Zimmermann have recently managed to construct a surjective group homomorphism from the high-dimensional Cremona group Cr n (k) to a free product of direct sums of Z/2Z, where n 3 and k ⊂ C is a subfield [3]. (More constructions in this setting can be found in [4,19].) For the high-dimensional case, it turned out that it is more suitable not to use the high-dimensional analogue of [13] but to take a different construction.…”

Section: Introductionmentioning

confidence: 99%

Relations in the Cremona group over a perfect field

Schneider¹

2022

Annales De l'Institut Fourier

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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.

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Quotients of groups of birational transformations of cubic del Pezzo fibrations

Abstract: Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.

Cited by 9 publications

References 26 publications

Rigid birational involutions of ℙ 3 and cubic threefolds

Rigid birational involutions of ℙ 3 and cubic threefolds

Properties of the Cremona group endowed with the Euclidean topology

Relations in the Cremona group over a perfect field

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