Length in the Cremona group
[Longueur dans le groupe de Cremona]
Annales Henri Lebesgue, Tome 2 (2019), pp. 187-257.

Le groupe de Cremona est le groupe des transformations birationnelles du plan. Une transformation birationnelle pour laquelle il existe un pinceau de droites envoyé sur un autre pinceau de droites est appelée transformation de Jonquières. D’après le célèbre théorème de Noether–Castelnuovo, chaque transformation birationnelle f est le produit de transformations de Jonquières. Le nombre minimal de facteurs dans un tel produit sera appelé la longueur de f et noté lgth(f). Même si cette longueur est assez imprévisible, nous donnons un algorithme permettant de la calculer, qui dépend seulement des multiplicités du système linéaire de f.

Eutre autres applications, nous donnons quelques propriétés de la longueur dynamique de f, définie comme étant la limite de la suite nlgth(f n )/n. Il s’ensuit par exemple qu’un élément du groupe de Cremona est distordu si et seulement s’il est algébrique. Le calcul de la longueur peut également être appliqué au complexe de Wright associé au groupe de Cremona : ceci a été fait récemment par Lonjou. De plus, nous montrons que la restriction de la longueur au groupe des automorphismes du plan affine est la longueur habituelle de ce dernier (la longueur donnée par la structure de produit amalgamé). Dans un autre ordre d’idées, nous calculons les longueurs et longueurs dynamiques de toutes les transformations monomiales et de certaines transformations de Halphen. Finalement, nous démontrons que la longueur est une application semi-continue inférieurement sur le groupe de Cremona, ce dernier étant muni de sa topologie de Zariski.

The Cremona group is the group of birational transformations of the plane. A birational transformation for which there exists a pencil of lines which is sent onto another pencil of lines is called a Jonquières transformation. By the famous Noether–Castelnuovo theorem, every birational transformation f is a product of Jonquières transformations. The minimal number of factors in such a product will be called the length, and written lgth(f). Even if this length is rather unpredictable, we provide an explicit algorithm to compute it, which only depends on the multiplicities of the linear system of f.

As an application of this computation, we give a few properties of the dynamical length of f defined as the limit of the sequence nlgth(f n )/n. It follows for example that an element of the Cremona group is distorted if and only if it is algebraic. The computation of the length may also be applied to the so called Wright complex associated with the Cremona group: This has been done recently by Lonjou. Moreover, we show that the restriction of the length to the automorphism group of the affine plane is the classical length of this latter group (the length coming from its amalgamated structure). In another direction, we compute the lengths and dynamical lengths of all monomial transformations, and of some Halphen transformations. Finally, we show that the length is a lower semicontinuous map on the Cremona group endowed with its Zariski topology.

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DOI : 10.5802/ahl.18
Classification : 14E07
Mots clés : Cremona transformations, Jonquières transformations, length, dynamical length, linear systems, base-points
Blanc, Jérémy 1 ; Furter, Jean-Philippe 2

1 Universität Basel Spiegelgasse 1 CH-4051 Basel (Switzerland)
2 Dpt. of Math., Univ. of La Rochelle avenue Crépeau 17000 La Rochelle (France)
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Blanc, Jérémy; Furter, Jean-Philippe. Length in the Cremona group. Annales Henri Lebesgue, Tome 2 (2019), pp. 187-257. doi : 10.5802/ahl.18. http://archive.numdam.org/articles/10.5802/ahl.18/

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