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The topological type of the Fano surface of a real three-dimensional \(M\)-cubic. (English. Russian original) Zbl 1106.14049
Izv. Math. 69, No. 6, 1137-1167 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 6, 61-94 (2005).
A real algebraic variety is called an \(M\)-variety if its total Betti number with \({\mathbb Z}/2\) coefficients is equal to the respective total Betti number of its complexification. The set of real straight lines on a real three-dimensional cubic variety is parameterized by a real Fano surface. In a previous work [Math. Notes 78, No. 5, 662–668 (2005; Zbl 1104.14018)] the author has shown that this Fano surface is an \(M\)-surface if and only if the cubic variety is an \(M\)-variety.
In the present paper it is shown that the real point set of the Fano \(M\)-surface, parameterizing the real lines of a real non-singular three-dimensional cubic \(M\)-variety, consists of 16 connected components, one homeomorphic to the connected sum of 5 real projective planes, and the others homeomorphic to the torus. The proof uses a degeneration of a non-singular cubic variety to a cubic with a singular double point.
14P25 Topology of real algebraic varieties
14D99 Families, fibrations in algebraic geometry
14J30 \(3\)-folds
14J45 Fano varieties
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