Length in the Cremona group

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 $\mathrm{lgth}\left(f\right)$. 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 $n\mapsto \mathrm{lgth}\left(f^n\right)/n$. 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 $\mathrm{lgth}\left(f\right)$. 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 $n\mapsto \mathrm{lgth}\left(f^n\right)/n$. 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.

[AC02] Alberich-Carramiñana, Maria Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769, Springer, 2002 | MR 1874328 | Zbl 0991.14008

[Ale16] Alexander, James W. On the factorization of Cremona plane transformations, Trans. Am. Math. Soc., Volume 17 (1916) no. 3, pp. 295-300 | Article | MR 1501043

[BC16a] Blanc, Jérémy; Calabri, Alberto On degenerations of plane Cremona transformations, Math. Z., Volume 282 (2016) no. 1-2, pp. 223-245 | Article | MR 3448381 | Zbl 1354.14024

[BC16b] Blanc, Jérémy; Cantat, Serge Dynamical degrees of birational transformations of projective surfaces, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 415-471 | Article | MR 3454379 | Zbl 1394.14011

[BCM15] Bisi, Cinzia; Calabri, Alberto; Mella, Massimiliano On plane Cremona transformations of fixed degree, J. Geom. Anal., Volume 25 (2015) no. 2, pp. 1108-1131 | Article | MR 3319964 | Zbl 1322.14034

[BD15] Blanc, Jérémy; Déserti, Julie Degree growth of birational maps of the plane, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 2, pp. 507-533 | MR 3410471 | Zbl 1342.14029

[BF13] Blanc, Jérémy; Furter, Jean-Philippe Topologies and structures of the Cremona groups, Ann. Math., Volume 178 (2013) no. 3, pp. 1173-1198 | Article | MR 3092478 | Zbl 1298.14020

[Bla11] Blanc, Jérémy Elements and cyclic subgroups of finite order of the Cremona group, Comment. Math. Helv., Volume 86 (2011) no. 2, pp. 469-497 | Article | MR 2775137 | Zbl 1213.14029

[Bla12] Blanc, Jérémy Simple Relations in the Cremona Group, Proc. Am. Math. Soc., Volume 140 (2012), pp. 1495-1500 | Article | MR 2869134 | Zbl 1251.14008

[Bla16] Blanc, Jérémy Conjugacy classes of special automorphisms of the affine spaces, Algebra Number Theory, Volume 10 (2016) no. 5, pp. 939-967 | Article | MR 3531358 | Zbl 1361.14032

[BvdPSZ14] Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim Neverending fractions. An introduction to continued fractions, Australian Mathematical Society Lecture Series, 23, Cambridge University Press, 2014, x+212 pages | Zbl 1307.11001

[Cas01] Castelnuovo, Guido Le trasformazioni generatrici del gruppo cremoniano nel piano, Torino Atti, Volume 36 (1901), pp. 861-874 | Zbl 32.0675.03

[CdC18] Cantat, Serge; de Cornulier, Yves Distortion in Cremona groups (2018) (https://arxiv.org/abs/1806.01674)

[Cor95] Corti, Alessio Factoring birational maps of threefolds after Sarkisov, J. Algebr. Geom., Volume 4 (1995) no. 2, pp. 223-254 | MR 1311348 | Zbl 0866.14007

[dC13] de Cornulier, Yves The Cremona group is not an amalgam, Acta Math., Volume 210 (2013) no. 1, pp. 31-94

[Dem70] Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 507-588 | Article | Numdam | MR 284446 | Zbl 0223.14009

[DF01] Diller, Jeffrey; Favre, Charles Dynamics of bimeromorphic maps of surfaces, Am. J. Math., Volume 123 (2001) no. 6, pp. 1135-1169 | Article | MR 1867314 | Zbl 1112.37308

[Dol12] Dolgachev, Igor V. Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | Zbl 1252.14001

[Fra49] Frame, James Sutherland Classroom Notes: Continued Fractions and Matrices, Am. Math. Mon., Volume 56 (1949) no. 2, pp. 98-103 | MR 1527170

[Fur02] Furter, Jean-Philippe On the length of polynomial automorphisms of the affine plane, Math. Ann., Volume 322 (2002) no. 2, pp. 401-411 | Article | MR 1893923 | Zbl 0996.14031

[Fur09] Furter, Jean-Philippe Plane polynomial automorphisms of fixed multidegree, Math. Ann., Volume 343 (2009) no. 4, pp. 901-920 | Article | MR 2471605 | Zbl 1187.14064

[Giz80] Gizatullin, Marat Kh. Rational $G$-surfaces, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 44 (1980) no. 1, pp. 110-144 | Zbl 0428.14022

[Giz82] Gizatullin, Marat Kh. Defining relations for the Cremona group of the plane, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 46 (1982), pp. 909-970 | MR 675525 | Zbl 0509.14011

[Har77] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | Zbl 0367.14001

[Har87] Harbourne, Brian Rational surfaces with infinite automorphism group and no antipluricanonical curve, Proc. Am. Math. Soc., Volume 99 (1987) no. 3, pp. 409-414 | Article | MR 875372 | Zbl 0643.14019

[Isk85] Iskovskikh, Vasiliĭ A. Proof of a theorem on relations in a two-dimensional Cremona group, Usp. Mat. Nauk, Volume 40 (1985) no. 5, pp. 255-256 English transl. in Russian Math. Surveys 40 (1985), no. 5, 231–232 | MR 810819 | Zbl 0613.14012

[Jun42] Jung, Heinrich Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., Volume 184 (1942), pp. 161-174 | MR 8915 | Zbl 0027.08503

[Lam01] Lamy, Stéphane L’alternative de Tits pour Aut[C2], J. Algebra, Volume 239 (2001) no. 2, pp. 413-437 | Article | MR 1832900 | Zbl 1040.37031

[Lam02] Lamy, Stéphane Une preuve géométrique du théorème de Jung, Enseign. Math., Volume 48 (2002) no. 3, pp. 3-4 | MR 1955604 | Zbl 1044.14035

[Lon18] Lonjou, Anne Graphes associés au groupe de Cremona (2018) (https://arxiv.org/abs/1802.02910)

[MFK94] Mumford, David B.; Fogarty, John; Kirwan, Frances C. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer, 1994, xiv+292 pages | MR 1304906

[Rui93] Ruiz, Jésus M. The basic theory of power series, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, 1993, x+134 pages

[Ser80] Serre, Jean-Pierre Trees, Springer, 1980, ix+142 pages (Translated from the French by John Stillwell) | Zbl 0548.20018

[Ser10] Serre, Jean-Pierre Le groupe de Cremona et ses sous-groupes finis, Séminaire Bourbaki. Volume 2008/2009 (Astérisque), Volume 332, Société Mathématique de France, 2010, pp. 75-100 | Numdam | Zbl 1257.14012

[vdK53] van der Kulk, Wouter On polynomial rings in two variables, Nieuw Arch. Wiskd., III. Ser., Volume 1 (1953), pp. 33-41 | MR 54574 | Zbl 0050.26002

[Wri92] Wright, David Two-dimensional Cremona groups acting on simplicial complexes, Trans. Am. Math. Soc., Volume 331 (1992) no. 1, pp. 281-300 | Article | MR 1038019 | Zbl 0767.14006