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 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.