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

Abstract : Let W -> X be a real smooth projective threefold fibred by rational curves. Kollár proved that if W(R) is orientable a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Let k : = k(N) be the integer defined as follows: If g : N -> F is a Seifert fibration, one defines k : = k(N) as the number of multiple fibres of g, while, if N is a connected sum of lens spaces, k is defined as the number of lens spaces different from P^3(R). Our Main Theorem says: If X is a geometrically rational surface, then k <= 4. Moreover we show that if F is diffeomorphic to S^1xS^1, then W(R) is connected and k = 0. These results answer in the affirmative two questions of Kollár who proved in 1999 that k <= 6 and suggested that 4 would be the sharp bound. We derive the Theorem from a careful study of real singular Del Pezzo surfaces with only Du Val singularities.
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Contributor : Frédéric Mangolte <>
Submitted on : Sunday, May 13, 2007 - 6:22:05 PM
Last modification on : Thursday, January 11, 2018 - 6:12:26 AM

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  • HAL Id : hal-00145949, version 1
  • ARXIV : 0705.0814



Fabrizio Catanese, Frédéric Mangolte. Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, I. Michigan Mathematical Journal, University of Michigan, 2008, 56, pp.357-373. ⟨hal-00145949⟩



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