Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Real algebraic morphisms and Del Pezzo surfaces of degree $2$

Authors: Nuria Joglar-Prieto and Frédéric Mangolte
Journal: J. Algebraic Geom. 13 (2004), 269-285
Published electronically: September 24, 2003
MathSciNet review: 2047699
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $X$ and $Y$ be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a smooth map $f:X\rightarrow Y$ can be approximated by regular maps in the space of ${\mathcal{C}}^\infty$ mappings from $X$ to $Y$, equipped with the ${\mathcal{C}}^\infty$ topology.

In this paper we give a complete solution to this problem when the target space is the standard 2-dimensional sphere and the source space is a geometrically rational real algebraic surface. The approximation result for real algebraic surfaces rational over $\mathbb R$ is due to J. Bochnak and W. Kucharz.

Here we give a detailed description of the more interesting case, namely real Del Pezzo surfaces of degree 2.

References [Enhancements On Off] (What's this?)

  • [BiMi] E. Bierstone and P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128, 207-302 (1997)
  • [BBK] J. Bochnak, M. Buchner and W. Kucharz, Vector bundles over real algebraic varieties, K-Theory 3, 271-298 (1989). Erratum, K-Theory 4, p. 103 (1990)
  • [BCR] J. Bochnak, M. Coste, M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. (3), vol. 12, Springer-Verlag, 1987; New edition: Real algebraic geometry, Ergeb. Math. Grenzgeb. (3), vol. 36, Springer-Verlag, 1998
  • [BKS] J. Bochnak, W. Kucharz, R. Silhol, Morphisms, line bundles and moduli spaces in real algebraic geometry Pub. Math. I.H.E.S. 86 (1997)
  • [DIK] A. Degtyarev, I. Itenberg, V. Kharlamov, Real Enriques Surfaces, Lecture Notes in Math. 1746, Springer-Verlag, 2000
  • [De] M. Demazure, Surfaces de Del Pezzo, II, III, IV et V, In: Séminaire sur les singularités des surface (Demazure, Pinkham, Teissier eds.), Lecture Notes in Math. 777, Springer-Verlag, 23-69 (1980)
  • [Ha] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag, 1977
  • [Hi] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. of Math. 79, 109-326 (1964)
  • [Jo] N. Joglar-Prieto, Rational surfaces and regular maps into the 2-dimensional sphere Math. Z. 234, 399-405 (2000)
  • [Ko] J. Kollár, Real algebraic surfacese-prints, alg-geom/9712003
  • [Ku] W. Kucharz, Algebraic morphisms into rational real algebraic surfaces J. Algebraic Geometry 8, 569-579 (1999)
  • [Ma] F. Mangolte, Cycles algébriques sur les surfaces K3 réelles, Math. Z. 225, 559-576 (1997)
  • [MR] F. Mangolte, C. Raffalli, Une question d'appuis, to appear (2001) http://www.lama.
  • [MS] J. Milnor, J. Stasheff, Characteristic classes, Princeton Univ. Press, Princeton, 1974
  • [Si] R. Silhol, Real Algebraic Surfaces, Lecture Notes in Math. 1392, Springer-Verlag, Berlin, 1989
  • [Ze] H. G. Zeuthen, Sur les différentes formes des courbes du quatrième ordre, Math. Ann. 7, 410-432 (1874)

Additional Information

Nuria Joglar-Prieto
Affiliation: ITIS CES Felipe II, Universidad Complutense de Madrid, C/Capitán 39, 28300 Aranjuez Madrid, Spain

Frédéric Mangolte
Affiliation: Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget du Lac Cedex, France

Received by editor(s): October 1, 2001
Published electronically: September 24, 2003
Additional Notes: The first author was supported by a Marie Curie Postdoctoral Fellowship (number HPMF-CT-1999-00019) at the Department of Mathematics at the Vrije Universiteit, Amsterdam

American Mathematical Society