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

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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 Faculté 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 \({\mathbb {R}}{\mathbb {P}}^{2}\).

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  1. Université de Nantes, Nantes, France

    Andrés Jaramillo Puentes

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Jaramillo Puentes, A. Rigid isotopy classification of generic rational curves of degree 5 in the real projective plane. Geom Dedicata 211, 1–70 (2021). https://doi.org/10.1007/s10711-020-00540-8

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