Rigid isotopy classification of generic rational quintics in $\mathbb{R}\mathbb{P}^{2}$

TL;DR

This paper classifies generic rational degree 5 curves in the real projective plane up to rigid isotopy by translating the problem into combinatorial dessins on surfaces, providing a new geometric-combinatorial approach.

Contribution

It introduces a novel method linking real rational quintic curves to dessins d'enfants on Hirzebruch surfaces for classification.

Findings

Complete rigid isotopy classification of generic rational quintics.

Development of combinatorial techniques for dessins decomposition.

Connection between dessins and curve isotopy classes.

Abstract

In this article we obtain the rigid isotopy classification of generic rational curves of degre $5$ in $\mathbb{R}\mathbb{P}^{2}$. In order to study the rigid isotopy classes of nodal rational curves of degree $5$ in $\mathbb{R}\mathbb{P}^{2}$, we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface $\Sigma_3$ and the corresponding nodal real dessin on $\mathbb{C}\mathbb{P}^{1}/(z\mapsto\bar{z})$. The dessins are real versions, proposed by S. Orevkov, of Grothendieck's dessins d'enfants. The dessins are graphs embedded in a topological surface and endowed with a certain additional structure. We study the combinatorial properties and decompositions of dessins corresponding to real nodal trigonal curves $C\subset \Sigma_n$ in real Hirzebruch surfaces $\Sigma_n$. Nodal dessins in the disk can be decomposed in blocks corresponding to cubic dessins in the disk $\mathbf{D}^2$, which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of real rational quintics in $\mathbb{R}\mathbb{P}^{2}$.