In order to study the rigid isotopy classes of nodal rational curves
of degree $5$ in $\mathbb{RP}^2$, we associate to every real rational
quintic curve with a marked real nodal point a trigonal curve in the
Hirzebruch surface $\Sigma_3$ and the corresponding nodal real dessin
on $\mathbb{CP}/(z\mapsto\bar{z})$. The dessins are real versions,
proposed by S. Orevkov, 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 $C\subset \Sigma$ in real ruled surfaces $\Sigma$. Uninodal
dessins in any surface with non-empty boundary and 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 $\mathbb{RP}^2$.
|
