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In this text, we study Viro’s conjecture and related problems for real algebraic surfaces in   $(\mathbb{CP}^1)^3$  . We construct a counterexample to Viro’s conjecture in tridegree   $(4,4,2)$   and a family of real algebraic surfaces of tridegree   $(d_1,d_2,2)$   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.

Real Algebraic Surfaces with Many Handles in $(\mathbb{CP}^1)^3$