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

TL;DR

This paper constructs specific real algebraic surfaces in $(\mathbb{CP}^1)^3$ that challenge existing conjectures and achieve high topological complexity, advancing understanding of real algebraic geometry.

Contribution

It provides a counter-example to Viro's conjecture and develops methods to construct surfaces with maximal Betti numbers in the given setting.

Findings

Counter-example to Viro's conjecture in tridegree (4,4,2)

Family of surfaces with asymptotically maximal first Betti number

Method for constructing surfaces using double covers and gluing singular curves

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.