A real sextic surface with 45 handles

TL;DR

This paper constructs a real nonsingular sextic surface in complex projective 3-space with 45 handles, surpassing previous known topological bounds using advanced algebraic geometry techniques.

Contribution

It introduces a new construction method for real sextic surfaces with high Betti numbers, improving the known maximum from prior results.

Findings

Constructed a real sextic with Betti number 90

Used Viro's patchworking and equivariant deformation techniques

Achieved a new topological bound for real sextic surfaces

Abstract

It follows from classical restrictions on the topology of real algebraic varieties that the first Betti number of the real part of a real nonsingular sextic in $\mathbb{CP}^3$ can not exceed $94$. We construct a real nonsingular sextic $X$ in $\mathbb{CP}^3$ satisfying $b_1(\mathbb{R}X)=90$, improving a result of F.Bihan. The construction uses Viro's patchworking and an equivariant version of a deformation due to E.Horikawa.