A real sextic surface with 45 handles

Abstract

It follows from classical restrictions on the topology of real algebraic varieties that the first Betti number of the real part of a real nonsingular sextic in ${\mathbb{CP}}^3$ can not exceed $94$. We construct a real nonsingular sextic $X$ in ${\mathbb{CP}}^3$ satisfying $b_1(\mathbb{R}X)=90$, improving a result of F.Bihan. The construction uses Viro's patchworking and an equivariant version of a deformation due to E.Horikawa.

18 pages, 13 figures