The group of Cremona transformations generated by linear maps and the standard involution
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2641-2680.

This article studies the group generated by automorphisms of the projective space of dimension n and by the standard birational involution of degree n. Every element of this group only contracts rational hypersurfaces, but in odd dimension, there are simple elements having this property which do not belong to the group. Geometric properties of the elements of the group are given, as well as a description of its intersection with monomial transformations.

Cet article étudie le groupe engendré par les automorphismes de l’espace projectif de dimension n et par l’involution birationnelle standard de degré n. Tout élément de ce groupe ne contracte que des hypersurfaces rationnelles, mais en dimension impaire il existe des éléments simples qui ont cette propriété et n’appartiennent pas au groupe. Des propriétés géométriques du groupe sont données, de même qu’une description de son intersection avec le groupe des transformations monômiales.

DOI: 10.5802/aif.2999
Classification: 14E07
Keywords: Cremona transformation, standard involution, rational hypersurfaces, monomial transformations
Mot clés : Transformations de Cremona, involution standard, hypersurfaces rationnelles, transformations monômiales
Blanc, Jérémy 1; Hedén, Isac 1

1 Mathematisches Institut Universität Basel Spiegelgasse 1 4051 Basel (Switzerland)
     author = {Blanc, J\'er\'emy and Hed\'en, Isac},
     title = {The group of {Cremona} transformations generated by linear maps and the standard involution},
     journal = {Annales de l'Institut Fourier},
     pages = {2641--2680},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2999},
     language = {en},
     url = {}
AU  - Blanc, Jérémy
AU  - Hedén, Isac
TI  - The group of Cremona transformations generated by linear maps and the standard involution
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 2641
EP  - 2680
VL  - 65
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  -
DO  - 10.5802/aif.2999
LA  - en
ID  - AIF_2015__65_6_2641_0
ER  - 
%0 Journal Article
%A Blanc, Jérémy
%A Hedén, Isac
%T The group of Cremona transformations generated by linear maps and the standard involution
%J Annales de l'Institut Fourier
%D 2015
%P 2641-2680
%V 65
%N 6
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.2999
%G en
%F AIF_2015__65_6_2641_0
Blanc, Jérémy; Hedén, Isac. The group of Cremona transformations generated by linear maps and the standard involution. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2641-2680. doi : 10.5802/aif.2999.

[1] Encyclopädie der mathematischen Wissenschaften. Band III, 2. Teil, 2 Häfte, Teubner, Leipzig (1921-1934)

[2] Alberich-Carramiñana, Maria Geometry of the plane Cremona maps, Lecture Notes in Mathematics, 1769, Springer-Verlag, Berlin, 2002, pp. xvi+257 | DOI | MR | Zbl

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

[4] Castelnuovo, Guido Le transformazioni generatrici del gruppo Cremoniano nel piano., Torino Atti, Volume 36 (1901), pp. 861-874

[5] Coble, Arthur B. Point sets and allied Cremona groups. II, Trans. Amer. Math. Soc., Volume 17 (1916) no. 3, pp. 345-385 | DOI | MR

[6] Déserti, Julie Some properties of the group of birational maps generated by the automorphisms of n and the standard involution (2014) (preprint, | MR

[7] Dolgachev, Igor; Ortland, David Point sets in projective spaces and theta functions, Astérisque (1988) no. 165, pp. 210 pp. (1989) | MR | Zbl

[8] Dolgachev, Igor V. Cremona special sets of points in products of projective spaces, Complex and differential geometry (Springer Proc. Math.), Volume 8, Springer, Heidelberg, 2011, pp. 115-134 | DOI | MR | Zbl

[9] Du Val, Patrick Application des idées cristallographiques à l’étude des groupes de transformations crémoniennes, 3 ième Coll. Géom. Algébrique (Bruxelles, 1959), Centre Belge Rech. Math., Louvain, 1960, pp. 65-73 | MR | Zbl

[10] Du Val, Patrick Crystallography and Cremona transformations, The geometric vein, Springer, New York-Berlin, 1981, pp. 191-201 | MR | Zbl

[11] van den Essen, Arno Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser Verlag, Basel, 2000, pp. xviii+329 | DOI | MR | Zbl

[12] Gizatullin, Marat On some tensor representations of the Cremona group of the projective plane, New trends in algebraic geometry (Warwick, 1996) (London Math. Soc. Lecture Note Ser.), Volume 264, Cambridge Univ. Press, Cambridge, 1999, pp. 111-150 | DOI | MR | Zbl

[13] Gonzalez-Sprinberg, Gerard; Pan, Ivan On characteristic classes of determinantal Cremona transformations, Math. Ann., Volume 335 (2006) no. 2, pp. 479-487 | DOI | MR | Zbl

[14] Hudson, Hilda Cremona transformations in plane and space, Cambridge, University Press, 1927, pp. XX + 454

[15] Huppert, B. Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967, pp. xii+793 | MR | Zbl

[16] Iskovskikh, V. A. Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi Mat. Nauk, Volume 51 (1996) no. 4(310), pp. 3-72 | DOI | MR | Zbl

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

[18] Kantor, S. Theorie der Transformationen Im R 3 , welche keine Fundamentalcurven 1. Art besitzen und ihrer endlichen gruppen, Acta Math., Volume 21 (1897) no. 1, pp. 1-78 | DOI | JFM | MR

[19] Kobb, Gustaf Sur la théorie des fonctions algébriques de deux variables., Journ. de Math. (4), Volume 8 (1892), pp. 385-419 | JFM

[20] van der Kulk, W. On polynomial rings in two variables, Nieuw Arch. Wiskunde (3), Volume 1 (1953), pp. 33-41 | MR | Zbl

[21] Marcolli, Matilde Feynman motives, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010, pp. xiv+220 | MR | Zbl

[22] Matsuki, Kenji Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002, pp. xxiv+478 | DOI | MR | Zbl

[23] Maubach, Stefan; Willems, Roel Polynomial automorphisms over finite fields: mimicking tame maps by the Derksen group, Serdica Math. J., Volume 37 (2011) no. 4, p. 305-322 (2012) | MR | Zbl

[24] Muir, Thomas A treatise of the theory of determinants with graduated sets of exercices, Macmillan, London, 1882 | JFM

[25] Muir, Thomas The theory of determinants in the historical order of development. Vol. III: The period 1861 to 1880, Macmillan, London, 1920

[26] Pan, Ivan Une remarque sur la génération du groupe de Cremona, Bol. Soc. Brasil. Mat. (N.S.), Volume 30 (1999) no. 1, pp. 95-98 | DOI | MR | Zbl

[27] Pascal, Ernesto Die Determinanten. Eine Darstellung ihrer Theorie und Anwendungen mit Rücksicht auf die neueren Forschungen, Teubner, Leipzig, 1900 | JFM

[28] Algebraic surfaces, 75 (1967)

[29] Semple, J. G.; Roth, L. Introduction to Algebraic Geometry, Oxford, at the Clarendon Press, 1949, pp. xvi+446 | MR | Zbl

[30] Shestakov, Ivan P.; Umirbaev, Ualbai U. The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc., Volume 17 (2004) no. 1, p. 197-227 (electronic) | DOI | MR | Zbl

[31] du Val, Patrick On the Directrices of a Set of Points in a Plane, Proc. London Math. Soc., Volume S2-35 (1933) no. 1, pp. 23 | DOI | MR | Zbl

Cited by Sources: