Real Algebraic and Analytic Geometry |

Real singular Del Pezzo surfaces and threefolds fibred by rational curves, I.

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Submission: 2007, July 25.

*Abstract:
Let W -> X be a real smooth projective threefold fibred by rational
curves. Koll\'ar 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 Koll\'ar 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.
(final version to appear in Michigan Mathematical Journal).*

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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