The main result of the paper under review is the following:
Theorem. Let \(X\) be a projective surface defined over \(\mathbb R\). Suppose that \(X\) is geometrically rational with Du Val singularities. Then, a connected component \(M\) of the topological normalization of the set \(X(\mathbb R)\) of real points of \(X\) contains at most four Du Val singular points that are either not of type \(A^-\) or of type \(A^-\) but globally separating.
Applying this theorem to rational curve fibrations over rational surfaces, the authors answer two questions proposed by J. Kollár [J. Math. Sci., New York 94, No. 1, 996–1020 (1999; Zbl 0964.14014)]. The first of these applications is the following:
Corollary. Let \(W\to X\) be a real smooth projective \(3\)-fold fibred by rational curves over a geometrically rational surface \(X\). Suppose that \(W(\mathbb R)\) is orientable, and let \(N\) be a connected component of \(W(\mathbb R)\). Then, \(N=N'\#^a\mathbb P^3(\mathbb R)\#^b(S^1\times S^2)\), and denote \(k(N)\) the number of lens spaces if \(N'\) is a sum of lens spaces, and let \(k(N)\) be the number of multiple fibers if \(N'\to F\) is a Seifert fibration. Then, \(k(N)\leq 4\).
The second application is the following one:
Corollary. Let \(W\to X\) be a real smooth projective \(3\)-fold fibred by rational curves over a geometrically rational surface \(X\). Suppose that the fibration is defined over \(\mathbb R\) and that \(W(\mathbb R)\) is orientable. Let \(N\) be a connected component of \(W(\mathbb R)\) that admits a Seifert fibration \(g:N\to S^1\times S^1\). Then, \(g\) has no multiple fibers. furthermore, \(X\) is rational ver \(\mathbb R\) and \(W(\mathbb R)\) is connected.