Real Algebraic and Analytic Geometry
Submission: 2008, November 5.
Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives sharp estimates on the number and the multiplicities of the Seifert fibres and on 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. The technique we use is to construct Seifert fibrations as projectivized tangent bundles of Du Val surfaces. ( to appear in Annales Scientifiques de l'Ecole Normale Supérieure) .
Mathematics Subject Classification (2000): 14P25, 14M20, 14J26.
Keywords and Phrases: Del Pezzo surface, rationally connected algebraic variety, Seifert manifold, Du Val surface.
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