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 $${\mathbb {R}}{\mathbb {P}}^{2}$$.

中文翻译：

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实射影平面中 5 次一般有理曲线的刚性同位素分类

在本文中，我们在 $${\mathbb {R}}{\mathbb {P}}^{2}$$ 中获得了 5 度泛型有理曲线的刚性同位素分类。为了研究 $${\mathbb {R}}{\mathbb {P}}^{2}$$ 中 5 次节点有理曲线的刚性同位素类，我们将每个实有理节点五次曲线与标记实节点点 Hirzebruch 曲面中的一个节点三角曲线 $$\Sigma_3$$ 和 $${\mathbb {C}}{\mathbb {P}}^{1}/(z\映射到 {\bar{z}})$$。dessins 是由 Orevkov (Annales de la Faculte des Sciences de Toulouse 12(4):517–531, 2003) 提出的格洛腾迪克的 dessins d'enfants 的真实版本。dessins 是嵌入在拓扑表面中并具有某种附加结构的图。我们研究了对应于真实 Hirzebruch 曲面 $$\Sigma _n$$ 中实节点三角曲线 $$C\subset\Sigma _n$$ 的 dessins 的组合属性和分解。磁盘中的节点dessins可以分解为与磁盘$${\mathbf {D}}^2$$中的三次dessins相对应的块，从而产生这些dessin的分类。正在考虑的 dessins 分类导致 $${\mathbb {R}}{\mathbb {P}}^{2}$$ 中实有理五次元的严格同位素分类。