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Sous-Groupes Algébriques du Groupe de Cremona

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Abstract

We give a complete classification of maximal algebraic subgroups of the Cremona group Bir(\( \mathbb{P}^{{\text{2}}} \)) and provide algebraic varieties that parametrize the conjugacy classes.

Résumé

Résumé Nous donnons une classification complète des sous-groupes algébriques maximaux du groupe de Cremona Bir(\( \mathbb{P}^{{\text{2}}} \)) et explicitons les variétés qui paramètrent les classes de conjugaison.

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Références

  1. F. Bars, On the Automorphisms Groups of Genus 3 curves, Notes del Seminari de Teoria Nombres UB-UAB-UPC, 2004/05, available at http://www.mat.uab.es/∼francesc/.

  2. L. Bayle, A. Beauville, Birational involutions of \( \mathbb{P}^{{\text{2}}} \), Asian J. Math. 4 (2000), no. 1, 11–17.

    MATH  MathSciNet  Google Scholar 

  3. A. Beauville, Complex Algebraic Surfaces, London Math. Soc. Student Texts, Vol. 34, Cambridge University Press, Cambridge, 1996.

    Google Scholar 

  4. A. Beauville, p-Elementary subgroups of the Cremona group, J. Algebra 314 (2007), no. 2, 553–564.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Beauville, J. Blanc, On Cremona transformations of prime order, C. R. Acad. Sci. Paris, Sér. I 339 (2004), 257–259.

    MATH  MathSciNet  Google Scholar 

  6. J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C. R. Acad. Sci. Paris, Sér. I 344 (2007), 21–26.

    MATH  MathSciNet  Google Scholar 

  7. J. Blanc, The number of conjugacy classes of elements of the Cremona group of some given finite order, Bull. Soc. Math. France 135 (2007), no. 3, 419–434.

    MATH  MathSciNet  Google Scholar 

  8. J. Blanc, Linearisation of finite abelian subgroups of the Cremona group of the plane, à paraître dans Groups Geom. Dyn., arXiv:0704.0537v1.

  9. A. Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223–254.

    MATH  MathSciNet  Google Scholar 

  10. T. de Fernex, On planar Cremona maps of prime order, Nagoya Math. J. 174 (2004), 1–28.

    MATH  MathSciNet  Google Scholar 

  11. M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. 3 (1970), 507–588.

    MATH  MathSciNet  Google Scholar 

  12. M. Demazure, Surfaces de Del Pezzo II, in: Séminaire sur les singularités des surfaces, Palaiseau, France, (1976–1977), Lecture Notes in Mathematics, Vol. 777, Springer-Verlag, New York, 22–70.

  13. I. V. Dolgachev, Weyl groups and Cremona transformations, in: Singularities I, Proc. Sympos. Pure Math., Vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 283–294.

  14. I. V. Dolgachev, Topics in Classical Algebraic Geometry, Part I, manuscript in preparation, available at http://www.math.lsa.umich.edu/~idolga/.

  15. I. V. Dolgachev, V. A. Iskovskikh, Finite subgroups of the plane Cremona group, à paraître dans Algebra, Arithmetic, and Geometry–Manin Festschrift, arXiv:math/0610595v2.

  16. F. Enriques, Sui gruppi continui di transformazioni cremoniani nel piano, Rend. Accad. Lincei, 1er sem., 1893.

  17. B. Harbourne, Rational surfaces with infinite automorphism group and no antipluricanonical curve, Proc. Amer. Math. Soc. 99 (1987), no. 3, 409–414.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), 651–677.

    Article  MATH  MathSciNet  Google Scholar 

  19. В. А. Исковских, МинималІные модели рационалІных поверхностей над произволІными полями, Изв. АН СССР, сер. мат. 43 (1979), no. 1, 19–43, 237. Engl. transl.: V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary fields, Math. USSR, Izv. 14 (1980), 17–39.

    Google Scholar 

  20. В. А. Исковских, Факторизация бирационалІных отображений рационалІных поверхностей с тояки зрения теории Мори, УМН 51 (1996), no. 4 (310), 3–72. English transl.: V. A. Iskovskikh, Factorization of birational map-pings of rational surfaces from the point of view of Mori theory, Russ. Math. Surv. 51 (1996), no.4, 585–652.

    Google Scholar 

  21. V. A. Iskovskikh, On finite subgroups of the Cremona group, Vietnam J. Math. 33 (2005), Special Issue, 61–80.

    MATH  MathSciNet  Google Scholar 

  22. S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Mayer & Müller, Berlin, 1895.

  23. J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 32, Springer-Verlag, Berlin, 1995.

  24. J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, Vol. 134, Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  25. J. Kollár, K. Smith, A. Corti, Rational and Nearly Rational Varieties, Cambridge Studies in Advanced Mathematics, Vol. 92, Cambridge University Press, Cambridge, 2004.

    Google Scholar 

  26. Ю. Манин, РационалІные поверхности над соверхенными полями ІІ, Мат. Сб., нов. сер. 72(114) (1967), 161–192. English transl.: Yu. Manin, Rational surfaces over perfect fields II, Math. USSR, Sb. 1 (1967), 141–168.

  27. S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), no. 1, 133–176.

    Article  MathSciNet  Google Scholar 

  28. T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Bd. 15. Springer-Verlag, Berlin, 1988.

  29. Э. Б. Винберг, В. Л. Попов, Теория инвариантов, Итоги науки и техники, Совр. пробл. матем., Фундам. направл., т. 55, М., ВИНИТИ, 1989, стр. 137–314. English transl.: V. L. Popov, E. B. Vinberg, Invariant theory, in: Algebraic Geometry, Vol. IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

  30. B. Segre, The Non-Singular Cubic Surfaces. Oxford University Press, Oxford, 1942.

    Google Scholar 

  31. J. P. Serre, Corps Locaux Deuxième édition, Publications de l'Université de Nancago, No. VIII, Hermann, Paris, 1968.

  32. H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.

    MATH  MathSciNet  Google Scholar 

  33. H. Umemura, Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables, Nagoya Math. J. 79 (1980), 47–67.

    MATH  MathSciNet  Google Scholar 

  34. H. Umemura, Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type, Nagoya Math. J. 87 (1982), 59–78.

    MATH  MathSciNet  Google Scholar 

  35. H. Umemura, On the maximal connected algebraic subgroups of the Cremona group, I, Nagoya Math. J. 88 (1982), 213–246.

    MATH  MathSciNet  Google Scholar 

  36. H. Umemura, On the maximal connected algebraic subgroups of the Cremona group, II, in: Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., Vol. 6, North-Holland, Amsterdam, 1985, pp. 349–436.

  37. M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443.

    Article  MATH  MathSciNet  Google Scholar 

  38. A. Weil, On algebraic groups of transformations, Amer. J. Math. 77 (1955), 355–391.

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Wiman, Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene, Math. Ann. 48 (1896), 195–240.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Université de Grenoble I, Institut Fourier, BP 74 38402, Saint-Martin d’Hères, France

    Jérémy Blanc

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Soutenu par le fonds national suisse de la recherche scientifique (FNRS).

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Blanc, J. Sous-Groupes Algébriques du Groupe de Cremona. Transformation Groups 14, 249–285 (2009). https://doi.org/10.1007/s00031-008-9046-5

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