Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

Fabrizio Catanese; Frédéric Mangolte

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 4, page 531-557
  • ISSN: 0012-9593

Abstract

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Let W → X be a real smooth projective 3-fold fibred by rational curves such that W ( ℝ ) is orientable. J. Kollár proved that a connected component N of W ( ℝ ) is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F , our result generalizes Comessatti’s theorem on smooth real rational surfaces: F cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti’s theorem, there are examples where F is non orientable, of hyperbolic type, and X is minimal.

How to cite

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Catanese, Fabrizio, and Mangolte, Frédéric. "Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II." Annales scientifiques de l'École Normale Supérieure 42.4 (2009): 531-557. <http://eudml.org/doc/272183>.

@article{Catanese2009,
abstract = {Let $W \rightarrow X$ be a real smooth projective 3-fold fibred by rational curves such that $W(\mathbb \{R\})$ is orientable. J. Kollár proved that a connected component $N$ of $W(\mathbb \{R\})$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$, our result generalizes Comessatti’s theorem on smooth real rational surfaces: $F$ cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti’s theorem, there are examples where $F$ is non orientable, of hyperbolic type, and $X$ is minimal.},
author = {Catanese, Fabrizio, Mangolte, Frédéric},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Del Pezzo surface; rationally connected algebraic variety; Seifert manifold; Du val surface},
language = {eng},
number = {4},
pages = {531-557},
publisher = {Société mathématique de France},
title = {Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II},
url = {http://eudml.org/doc/272183},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Catanese, Fabrizio
AU - Mangolte, Frédéric
TI - Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 4
SP - 531
EP - 557
AB - Let $W \rightarrow X$ be a real smooth projective 3-fold fibred by rational curves such that $W(\mathbb {R})$ is orientable. J. Kollár proved that a connected component $N$ of $W(\mathbb {R})$ is essentially either Seifert fibred or a connected sum of lens spaces. Answering three questions of Kollár, we give sharp estimates on the number and the multiplicities of the Seifert fibres (resp. the number and the torsions of the lens spaces) when $X$ is a geometrically rational surface. When $N$ is Seifert fibred over a base orbifold $F$, our result generalizes Comessatti’s theorem on smooth real rational surfaces: $F$ cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti’s theorem, there are examples where $F$ is non orientable, of hyperbolic type, and $X$ is minimal.
LA - eng
KW - Del Pezzo surface; rationally connected algebraic variety; Seifert manifold; Du val surface
UR - http://eudml.org/doc/272183
ER -

References

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  9. [9] F. Mangolte, Cycles algébriques sur les surfaces K 3 réelles, Math. Z.225 (1997), 559–576. Zbl0914.14019MR1466402
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  14. [14] R. Silhol, Real algebraic surfaces, Lecture Notes in Math. 1392, Springer, 1989. Zbl0691.14010MR1015720

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