Real algebraic surfaces with many handles in $(\mathbb{CP}^1)^3$

Abstract

In this text, we study Viro's conjecture and related problems for real algebraic surfaces in $(\mathbb{CP}^1)^3$. We construct a counter-example to Viro's conjecture in tridegree $(4,4,2)$ and a family of real algebraic surfaces of tridegree $(2k,2l,2)$ in $(\mathbb{CP}^1)^3$ with asymptotically maximal first Betti number of the real part. To perform such constructions, we consider double covers of blow-ups of $(\mathbb{CP}^1)^2$ and we glue singular curves with special position of the singularities adapting the proof of Shustin's theorem for gluing singular hypersurfaces.