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 RP 2 . In order to study the rigid isotopy classes of nodal rational curves of degree 5 in RP 2 , we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface Σ 3 and the corresponding nodal real dessin on CP 1 /(z → z). The dessins are real versions, proposed by S. Orevkov [10], 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 ⊂ Σn in real Hirzebruch surfaces Σn. Nodal dessins in the disk can be decomposed in blocks corresponding to cubic dessins in the disk 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 RP 2 . Contents 42 3.1. Hirzebruch surfaces 42 3.2. Positive and negative inner nodal ×-vertices 43 4. Generic quintic rational curves in RP 2 47 4.1. Maximally perturbable curves 50 4.2. (M − 2)-perturbable curves 55 4.3. (M − 4)-perturbable curves 63 4.4. (M − 6)-perturbable curves 65 References 66