Metadata
Abstract
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 is a product of Jonquières transformations. The minimal number of factors in such a product will be called the length, and written . 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 .
As an application of this computation, we give a few properties of the dynamical length of defined as the limit of the sequence . 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.
References
[AC02] Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769, Springer, 2002 | MR | Zbl
[Ale16] On the factorization of Cremona plane transformations, Trans. Am. Math. Soc., Volume 17 (1916) no. 3, pp. 295-300 | DOI | MR
[BC16a] On degenerations of plane Cremona transformations, Math. Z., Volume 282 (2016) no. 1-2, pp. 223-245 | DOI | MR | Zbl
[BC16b] Dynamical degrees of birational transformations of projective surfaces, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 415-471 | DOI | MR | Zbl
[BCM15] On plane Cremona transformations of fixed degree, J. Geom. Anal., Volume 25 (2015) no. 2, pp. 1108-1131 | DOI | MR | Zbl
[BD15] Degree growth of birational maps of the plane, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 2, pp. 507-533 | MR | Zbl
[BF13] Topologies and structures of the Cremona groups, Ann. Math., Volume 178 (2013) no. 3, pp. 1173-1198 | DOI | MR | Zbl
[Bla11] Elements and cyclic subgroups of finite order of the Cremona group, Comment. Math. Helv., Volume 86 (2011) no. 2, pp. 469-497 | DOI | MR | Zbl
[Bla12] Simple Relations in the Cremona Group, Proc. Am. Math. Soc., Volume 140 (2012), pp. 1495-1500 | DOI | MR | Zbl
[Bla16] Conjugacy classes of special automorphisms of the affine spaces, Algebra Number Theory, Volume 10 (2016) no. 5, pp. 939-967 | DOI | MR | Zbl
[BvdPSZ14] Neverending fractions. An introduction to continued fractions, Australian Mathematical Society Lecture Series, 23, Cambridge University Press, 2014, x+212 pages | Zbl
[Cas01] Le trasformazioni generatrici del gruppo cremoniano nel piano, Torino Atti, Volume 36 (1901), pp. 861-874 | Zbl
[CdC18] Distortion in Cremona groups (2018) (https://arxiv.org/abs/1806.01674)
[Cor95] Factoring birational maps of threefolds after Sarkisov, J. Algebr. Geom., Volume 4 (1995) no. 2, pp. 223-254 | MR | Zbl
[dC13] The Cremona group is not an amalgam, Acta Math., Volume 210 (2013) no. 1, pp. 31-94
[Dem70] Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 507-588 | DOI | Numdam | MR | Zbl
[DF01] Dynamics of bimeromorphic maps of surfaces, Am. J. Math., Volume 123 (2001) no. 6, pp. 1135-1169 | DOI | MR | Zbl
[Dol12] Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | Zbl
[Fra49] Classroom Notes: Continued Fractions and Matrices, Am. Math. Mon., Volume 56 (1949) no. 2, pp. 98-103 | MR
[Fur02] On the length of polynomial automorphisms of the affine plane, Math. Ann., Volume 322 (2002) no. 2, pp. 401-411 | DOI | MR | Zbl
[Fur09] Plane polynomial automorphisms of fixed multidegree, Math. Ann., Volume 343 (2009) no. 4, pp. 901-920 | DOI | MR | Zbl
[Giz80] Rational -surfaces, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 44 (1980) no. 1, pp. 110-144 | Zbl
[Giz82] Defining relations for the Cremona group of the plane, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 46 (1982), pp. 909-970 | MR | Zbl
[Har77] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | Zbl
[Har87] Rational surfaces with infinite automorphism group and no antipluricanonical curve, Proc. Am. Math. Soc., Volume 99 (1987) no. 3, pp. 409-414 | DOI | MR | Zbl
[Isk85] Proof of a theorem on relations in a two-dimensional Cremona group, Usp. Mat. Nauk, Volume 40 (1985) no. 5, p. 255-256 English transl. in Russian Math. Surveys 40 (1985), no. 5, 231–232 | MR | Zbl
[Jun42] Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., Volume 184 (1942), pp. 161-174 | MR | Zbl
[Lam01] L’alternative de Tits pour Aut[C2], J. Algebra, Volume 239 (2001) no. 2, pp. 413-437 | DOI | MR | Zbl
[Lam02] Une preuve géométrique du théorème de Jung, Enseign. Math., Volume 48 (2002) no. 3, p. 3-4 | MR | Zbl
[Lon18] Graphes associés au groupe de Cremona (2018) (https://arxiv.org/abs/1802.02910)
[MFK94] Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34, Springer, 1994, xiv+292 pages | MR
[Rui93] The basic theory of power series, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, 1993, x+134 pages
[Ser80] Trees, Springer, 1980, ix+142 pages (Translated from the French by John Stillwell) | Zbl
[Ser10] 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
[vdK53] On polynomial rings in two variables, Nieuw Arch. Wiskd., III. Ser., Volume 1 (1953), pp. 33-41 | MR | Zbl
[Wri92] Two-dimensional Cremona groups acting on simplicial complexes, Trans. Am. Math. Soc., Volume 331 (1992) no. 1, pp. 281-300 | DOI | MR | Zbl