# Seminar in Real and Complex Geometry

## Rigid isotopies of degree 5 nodal rational curves in \mathbb{RP}^2

Abstract

 In order to study the rigid isotopy classes of nodal rational curves of degree $5$ in $\mathbb{RP}^2$, we associate to every real rational quintic curve with a marked real nodal point a trigonal curve in the Hirzebruch surface $\Sigma_3$ and the corresponding nodal real dessin on $\mathbb{CP}/(z\mapsto\bar{z})$. 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 $C\subset \Sigma$ in real ruled surfaces $\Sigma$. Uninodal dessins in any surface with non-empty boundary and 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 $\mathbb{RP}^2$.