Betti numbers of real numerical quintic surfaces (Q2735109)

A real numerically quintic surface is a proper connected smooth real algebraic surface having the same numerical invariants as a smooth real quintic surface in \({\mathbb P}^3\), i.e, having \(p_g=4\), \(q=0\) and \(c_1^2=5\). Let \(X\) be a real numerically quintic surface. Let \(b_0\) (respectively \(b_1\)) be the \(0\)-th (respectively first) \({\mathbb Z}/2{\mathbb Z}\)-Betti numbers of the set \(X({\mathbb R})\) of real points of \(X\). Classical results imply that \(b_0\leq 25\) and that \(b_1\leq 47\). NEWLINENEWLINENEWLINEUsing Viro's method as described by \textit{J.-J. Risler} [Séminaire Bourbaki 1992/93, Astérisque 216, 69-86 (Exp. No. 763) (1993; Zbl 0824.14045)] and a result of \textit{E. Horikawa} [Invent. Math. 31, 43-85 (1975; Zbl 0317.14018)], the author shows that there are real numerically quintic surfaces \(X\) having \(b_1=47\). He also shows that there are real numerically quintic surfaces \(X\) having \(b_0=23\). A recent result of \textit{S. Yu. Orevkov} states that one may suppress the adverb ``numerically'' in the last sentence [C. R. Acad. Sci., Paris, Sér. I, Math. 333, 115-118 (2001; Zbl 1011.14017)].NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].