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

In this article we obtain the rigid isotopy classification of generic rational curves of degre 5 in RP2. In order to study the rigid isotopy classes of nodal rational curves of degree 5 in RP2, 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 CP1/( z -> (z) over bar). 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 of 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 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 RP2.