Rigid isotopy classification of generic rational curves of degree 5 in the real projective plane

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 Orevkov (Annales de la Faculte des sciences de Toulouse 12(4):517–531, 2003), 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 $${\mathbb {R}}{\mathbb {P}}^{2}$$ .