×

On birational transformations of pairs in the complex plane. (English) Zbl 1170.14010

The authors consider pairs \((S,C)\) where \(S\) is a rational surface and \(C\subset S\) is an irreducible curve. They say that a birational map \(\varphi:S\dashrightarrow S^{\prime}\) is a birational transformation of pairs \(\varphi:(S,C)\dashrightarrow (S^{\prime},C^{\prime})\) if it restricts to a birational transformation \(\varphi|_{C}:C\dashrightarrow C^{\prime}\). The group of birational transformations of a pair \((S,C)\) is denoted by \(\text{Dec}(S,C)\) and induces the so-called canonical complex \[ 1\rightarrow \text{Ine}(S,C)\rightarrow \text{Dec}(S,C) @>\rho>> \text{Bir}(C)\rightarrow 1 \] where \(\rho\) is the action of \(\text{Dec}(S,C)\) on \(C\) and \(\text{Ine}(S,C)\) is the inertia group of \(C\) in \(\text{Bir}(S)\) which is the group of birational transformations that fix \(C\). In the paper under review it is given a survey about the pairs \((\mathbb P ^2,C)\) whose decomposition group \(\text{Dec}(\mathbb P ^2,C)\) is not trivial, together with a description of their corresponding canonical complexes. The cases of curves of genus \(\geq 2\), 1 and 0 are considered separately. Moreover, it is expressed the link between the transformations that preserve or fix curves with respectively the classification of finite subgroups and the dynamics of the elements of \(\text{Bir}(\mathbb P ^2)\).

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14J26 Rational and ruled surfaces
14H50 Plane and space curves
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bayle L., Beauville A.: Birational involutions of \({\mathbb {P}^2}\) . Asian. J. Math. 4(1), 11–17 (2000) · Zbl 1055.14012
[2] Beauville, A.: Complex algebraic surfaces, London Math. Soc. Student Texts, 34 (1996) · Zbl 0849.14014
[3] Beauville A., Blanc J.: On Cremona transformations of prime order. C.R. Acad. Sci., Ser. 1, Math. 339(4), 257–259 (2004) · Zbl 1062.14017
[4] Bedford E., Diller J.: Energy and invariant measures for birational surface maps. Duke Math. J. 128(2), 331–368 (2005) · Zbl 1076.37031 · doi:10.1215/S0012-7094-04-12824-6
[5] Bedford, E., Kim, K.: Dynamics of rational surface automorphisms: linear fractional recurrences, arXiv:math/0611297 · Zbl 1185.37128
[6] Blanc J.: Finite Abelian subgroups of the Cremona group of the plane. C.R. Acad. Sci. Paris Ser. I, Math. 344, 21–26 (2007) · Zbl 1111.14003
[7] Blanc J.: The number of conjugacy classes of elements of the Cremona group of some given finite order. Bull. Soc. Math. France 135(3), 419–434 (2007) · Zbl 1158.14017
[8] Blanc J.: On the inertia group of elliptic curves in the Cremona group of the plane. Michigan Math. J. 56(2), 315–330 (2008) · Zbl 1156.14011 · doi:10.1307/mmj/1224783516
[9] Blanc, J.: Linearisation of finite Abelian subgroups of the Cremona group of the plane. Groups Geom. Dyn. arXiv:math/0704.0537 (to appear)
[10] Blanc J., Pan I., Vust T.: Sur un théorème de Castelnuovo. Bull. Braz. Math. Soc. (N.S.) 39(1), 61–80 (2008) · Zbl 1139.14014 · doi:10.1007/s00574-008-0072-7
[11] Castelnuovo, G.: Sulle trasformazioni cremoniane del piano che ammettono una curva fissa, Rend. Accad. Lincei (1892); Memorie scelte, Bologna, Zanichelli (1937)
[12] Coble, A.B.: Algebraic geometry and theta functions, Amer. Math. Soc. Colloquium Publications, Vol. X (1961) · JFM 55.0808.02
[13] Coolidge, J.L.,: A treatise on algebraic curves, Dover Publications, Inc. (1959) · Zbl 0085.36403
[14] de Fernex T.: On planar Cremona maps of prime order. Nagoya Math. J. 174, 1–28 (2004) · Zbl 1062.14019
[15] de Jonquières E.: De la transformation géométrique des figures planes, et d’un mode de génération de certaines courbes à double courbure de tous les ordres. Nouv. Ann. (2)3, 97–111 (1864)
[16] Diller J., Favre C.: Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123(6), 1135–1169 (2001) · Zbl 1112.37308 · doi:10.1353/ajm.2001.0038
[17] Diller J., Jackson D., Sommese A.: Invariant curves for birational surface maps. Trans. Amer. Math. Soc. 359(6), 2793–2991 (2007) · Zbl 1115.14007 · doi:10.1090/S0002-9947-07-04162-1
[18] Dolgachev, I.V., Iskovskikh, V.A.: Finite subgroups of the plane Cremona group. In: Arithmetic and Geometry-Manin Festschrift. math.AG/0610595 (2006)
[19] Dolgachev, I.V., Ortland, D.: Point sets in projective spaces and theta functions, Astérisque, No. 165 (1988), 210 pp. (1989) · Zbl 0685.14029
[20] Dujardin R.: Laminar currents and birational dynamics. Duke Math. J. 131(2), 219–247 (2006) · Zbl 1099.37037 · doi:10.1215/S0012-7094-06-13122-8
[21] Friedland, S.: Entropy of algebraic maps. Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl. Special Issue, 215–228 (1995) · Zbl 0890.54018
[22] Gizatullin M.H.: Rational G-surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 110–144, 239 (1980) · Zbl 0428.14022
[23] Gizatullin M.H.: The decomposition, inertia and ramification groups in birational geometry. Algebraic Geometry and its Applications, Aspects of Mathematics, E 25, 39–45 (1994) · Zbl 0834.14006
[24] Godeaux, L.: Géométrie algébrique II, géométrie sur une courbe algébrique, géométrie algébrique du plan, Masson & Cie, éditeurs, Paris (1950)
[25] Halphen M.: Sur les courbes planes du sixième degré à neuf points doubles. Bull. Soc. Math. France 10, 162–172 (1882) · JFM 14.0580.02
[26] Harbourne B.: Rational surfaces with infinite automorphism group and no antipluricanonical curve. Proc. AMS. 99(3), 409–414 (1987) · Zbl 0643.14019 · doi:10.1090/S0002-9939-1987-0875372-8
[27] Hudson, H.: Cremona transformations in plane and space, Cambridge University Press (1927) · JFM 53.0595.01
[28] Kollár, J., Smith, K., Corti, A.: Rational and nearly rational varieties. Cambridge Studies in Advanced Mathematics, 92 (2004) · Zbl 1060.14073
[29] Kumar M., Murthy P.: Curves with negative self-intersection on rational surfaces. J. Math. Kyoto Univ. 22(4), 767–777 (1982/83) · Zbl 0509.14033
[30] Küpper, C.: Ueber das Vorkommen von linearen Scharen \({g_n^{(2)}}\) auf Curven n ter Ordnung \({C_p^n}\) , deren Geschlecht p grösser ist als p 1, das Maximalgeschlecht einer Raumcurve \({\mathrm{\mathbf{R}}}_{p_1}^n\) , Prag. Ber. 264–272 (1892) · JFM 24.0664.01
[31] McMullen C.: Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes. Études Sci. 105, 49–89 (2007) · Zbl 1143.37033
[32] Nagata M.: On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32, 351–370 (1960) · Zbl 0100.16703
[33] Pan I.: Sur le degré dynamique des transformations de Cremona du plan qui stabilisent une courbe irrationnelle non-elliptique. C.R. Acad. Sci. Paris, Ser. I 341(7), 439–443 (2005) · Zbl 1080.37052
[34] Pan I.: Sur le sous-groupe de décomposition d’une courbe irrationnelle dans le groupe de Cremona du plan. Michigan Math. J. 55(2), 285–298 (2007) · Zbl 1144.14010 · doi:10.1307/mmj/1187646995
[35] Russakovskii A., Shiffman B.: Value distribution for sequences of rational mappings and complex dynamics. Ind. Univ. Math. J. 46, 897–932 (1997) · Zbl 0901.58023 · doi:10.1512/iumj.1997.46.1441
[36] Semple J.G., Roth L.: Introduction to algebraic geometry. Clarendon Press, Oxford (1949) · Zbl 0041.27903
[37] Snyder, V., Coble, A., Emch, A., Lefschetz, S., Sharp F., Sizam, C.: Selected topics in algebraic geometry, Report of the committee on rational transformations, Nat. Research Council, Chelsea Pub. Company (1970)
[38] Wiman A.: Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene. Math. Ann. 48, 195–240 (1896) · JFM 30.0600.01 · doi:10.1007/BF01446342
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.