Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 4, p. 531-557

Let WX be a real smooth projective 3-fold fibred by rational curves such that W() is orientable. J. Kollár proved that a connected component N of W() 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.

Soit WX une variété projective réelle non singulière munie d’une fibration en courbes rationnelles et telle que W() soit orientable. J. Kollár a montré qu’une composante connexe N de W() est essentiellement une variété de Seifert ou une somme connexe d’espaces lenticulaires. Répondant à trois questions de Kollár, nous donnons une estimation optimale du nombre et des multiplicités des fibres de Seifert (resp. du nombre et des torsions des espaces lenticulaires) lorsque X est une surface géométriquement rationnelle. Lorsque N admet une fibration de Seifert au-dessus d’un orbifold F, nos résultats généralisent le théorème de Comessatti sur les surfaces rationnelles réelles lisses  : F ne peut pas être à la fois orientable et de type hyperbolique. Nous montrons, ce qui est une surprise, qu’à la différence du théorème de Comessatti, il existe des exemples où F est non orientable, de type hyperbolique, et X est minimale.

DOI : https://doi.org/10.24033/asens.2102
Classification:  14P25,  14M20,  14J26
Keywords: Del Pezzo surface, rationally connected algebraic variety, Seifert manifold, Du val surface
@article{ASENS_2009_4_42_4_531_0,
     author = {Catanese, Fabrizio and Mangolte, Fr\'ed\'eric},
     title = {Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {4},
     year = {2009},
     pages = {531-557},
     doi = {10.24033/asens.2102},
     zbl = {1183.14075},
     mrnumber = {2568875},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_4_531_0}
}
Catanese, Fabrizio; Mangolte, Frédéric. Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 4, pp. 531-557. doi : 10.24033/asens.2102. http://www.numdam.org/item/ASENS_2009_4_42_4_531_0/

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