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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) isessentially 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 Seifertfibration, 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 lensspaces different from P^3(R). Our Main Theorem says: If X is a geometricallyrational surface, then k <= 4. Moreover we show that if F is diffeomorphic toS^1xS^1, then W(R) is connected and k = 0. These results answer in theaffirmative two questions of Koll\'ar who proved in 1999 that k <= 6 andsuggested that 4 would be the sharp bound. We derive the Theorem from a carefulstudy of real singular Del Pezzo surfaces with only Du Val singularities.

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