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.

arXiv comment29 pages, 8 figures