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arXiv · 1303.4341

Totally real pencils of cubics with respect to sextics

Séverine Fiedler-Le Touzé
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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.

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Séverine Fiedler-Le Touzé. 2013-03-18. Totally real pencils of cubics with respect to sextics. https://arxiv.org/abs/1303.4341

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