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.
[BvdPSZ14] Neverending fractions. An introduction to continued fractions, Australian Mathematical Society Lecture Series, Volume 23, Cambridge University Press, 2014, x+212 pages | Zbl 1307.11001
[Cas01] Le trasformazioni generatrici del gruppo cremoniano nel piano, Torino Atti, Volume 36 (1901), pp. 861-874 | Zbl 32.0675.03
[CdC18] Distortion in Cremona groups (2018) (https://arxiv.org/abs/1806.01674)
[dC13] The Cremona group is not an amalgam, Acta Math., Volume 210 (2013) no. 1, pp. 31-94
[Dol12] Classical algebraic geometry. A modern view, Cambridge University Press, 2012, xii+639 pages | Zbl 1252.14001
[Fra49] Classroom Notes: Continued Fractions and Matrices, Am. Math. Mon., Volume 56 (1949) no. 2, pp. 98-103 | MR 1527170
[Giz80] Rational -surfaces, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 44 (1980) no. 1, pp. 110-144 | Zbl 0428.14022
[Har77] Algebraic geometry, Graduate Texts in Mathematics, Volume 52, Springer, 1977, xvi+496 pages | Zbl 0367.14001
[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 810819 | Zbl 0613.14012
[Lon18] Graphes associés au groupe de Cremona (2018) (https://arxiv.org/abs/1802.02910)
[MFK94] Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 34, Springer, 1994, xiv+292 pages | MR 1304906
[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 0548.20018