Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-29T07:34:13.975Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometrically rational real conic bundles and very transitive actions

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  13 September 2010

Jérémy Blanc  and
Frédéric Mangolte
Jérémy Blanc
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Schweiz (email: Jeremy.Blanc@unibas.ch)
Frédéric Mangolte
Affiliation:
Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget du Lac Cedex, France (email: mangolte@univ-savoie.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

Keywords

real algebraic surfacesrational surfacesgeometrically rational surfacesbirational geometryalgebraic automorphismsvery transitive actionsCremona transformations

MSC classification

Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 14P25: Topology of real algebraic varieties 14J26: Rational and ruled surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 161 - 187
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Biswas, I. and Huisman, J., Rational real algebraic models of topological surfaces, Doc. Math. 12 (2007), 549–567.CrossRefGoogle Scholar
[2]Blanc, J., Linearisation of finite Abelian subgroups of the Cremona group of the plane, Groups Geom. Dyn. 3 (2009), 215–266.CrossRefGoogle Scholar
[3]Blanc, J., Sous-groupes algébriques du groupe de Cremona, Transform. Groups 14 (2009), 249–285.CrossRefGoogle Scholar
[4]Bochnak, J., Coste, M. and Roy, M.-F., Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), vol. 36 (Springer, Berlin, 1998).CrossRefGoogle Scholar
[5]Comessatti, A., Fondamenti per la geometria sopra superfizie razionali dal punto di vista reale, Math. Ann. 73 (1912), 1–72.CrossRefGoogle Scholar
[6]Comessatti, A., Sulla connessione delle superfizie razionali reali, Ann. Math. 23 (1914), 215–283.Google Scholar
[7]Demazure, M., Surfaces de Del Pezzo II. Séminaire sur les singularités des surfaces, Palaiseau, France, 1976–1977, Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 22–70.CrossRefGoogle Scholar
[8]Dolgachev, I. V. and Iskovskikh, V. A., Finite subgroups of the plane Cremona group, in Algebra, arithmetic and geometry, Vol. 1: in honor of Yu. I. Manin, Progress in Mathematics, vol. 269 (Birkhäuser, Boston, MA, 2009), 443–548.CrossRefGoogle Scholar
[9]Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid., 205–326.CrossRefGoogle Scholar
[10]Huisman, J. and Mangolte, F., The group of automorphisms of a real rational surface isn-transitive, Bull. Lond. Math. Soc. 41 (2009), 563–568.CrossRefGoogle Scholar
[11]Huisman, J. and Mangolte, F., Automorphisms of real rational surfaces and weighted blow-up singularities, Manuscripta Math. 132 (2010), 1–17.CrossRefGoogle Scholar
[12]Iskovskikh, V. A., Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 19–43 237.Google Scholar
[13]Iskovskikh, V. A., Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi Mat. Nauk 51 (1996), 3–72.Google Scholar
[14]Kollár, J., Real algebraic surfaces, arXiv:alg-geom/9712003v1.Google Scholar
[15]Kollár, J., The topology of real algebraic varieties, in Current developments in mathematics 2000 (International Press, Somerville, MA, 2001), 197–231.Google Scholar
[16]Kollár, J. and Mangolte, F., Cremona transformations and diffeomorphisms of surfaces, Adv. Math. 222 (2009), 44–61.CrossRefGoogle Scholar
[17]Mangolte, F., Real algebraic morphisms on 2-dimensional conic bundles, Adv. Geom. 6 (2006), 199–213.CrossRefGoogle Scholar
[18]Manin, Yu., Rational surfaces over perfect fields, II, Math. USSR - Sbornik 1 (1967), 141–168.CrossRefGoogle Scholar
[19]Ronga, F. and Vust, T., Diffeomorfismi birazionali del piano proiettivo reale, Comm. Math. Helv. 80 (2005), 517–540.Google Scholar
[20]Silhol, R., Real algebraic surfaces, Lecture Notes in Mathematics, vol. 1392 (Springer, Berlin, 1989).CrossRefGoogle Scholar
[21]Tognoli, A., Su una congettura di Nash, Ann. Sc. Norm. Super. Pisa (3) 27 (1973), 167–185.Google Scholar
[22]Ueno, K., Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, vol. 439 (Springer, Berlin, 1975).CrossRefGoogle Scholar