Totally real pencils of cubics with respect to sextics

TL;DR

This paper investigates whether certain dividing sextic curves with specific oval configurations can be equipped with totally real pencils of cubics, extending understanding of real algebraic curves and their intersection properties.

Contribution

It proves that dividing $M-2$-sextics with 2 or 6 empty exterior ovals can be endowed with totally real pencils of cubics, a new result in real algebraic geometry.

Findings

Dividing $M-2$-sextics with specified oval configurations admit totally real cubics.

Such sextics are always dividing.

The existence of totally real pencils is established for these curves.

Abstract

A real algebraic plane curve $A$ is said to be dividing if its real part $\mathbb{R}A$ disconnects its complex part $\mathbb{C}A$. A pencil of curves is totally real with respect to $A$ if it has only real intersections with $\mathbb{C}A$. If there exists such a pencil, then $A$ is dividing, this is the case for the $M$-curves. Can conversely any dividing curve be endowed with a totally real pencil? We study here the case of $M-2$-sextics having 2 or 6 empty exterior ovals. Such sextics are always dividing. We prove that they may actually be endowed with a totally real pencil of cubics.