Generic pointed quartic curves in $\mathbb{R}\mathbb{P}^{2}$ and uninodal dessins

TL;DR

This paper classifies generic pointed quartic curves in real projective plane using combinatorial properties of dessins d'enfants, providing a detailed rigid isotopy classification based on geometric and topological features.

Contribution

It introduces a new combinatorial approach using dessins to classify generic pointed quartic curves, extending previous work and providing explicit criteria for classification.

Findings

20 classes distinguished by oval count and tangent line interactions

Decomposition of dessins into cubic blocks for classification

Connection to previous classifications via del Pezzo surfaces

Abstract

In this article we obtain a rigid isotopy classification of generic pointed quartic curves $(A,p)$ in $\mathbb{R}\mathbb{P}^{2}$ by studying the combinatorial properties of dessins. The dessins are real versions, proposed by S. Orevkov, of Grothendieck's dessins d'enfants. This classification contains 20 classes determined by the number of ovals of $A$, the parity of the oval containing the marked point $p$, the number of ovals that the tangent line $T_p A$ intersects, the nature of connected components of $A\setminus T_p A$ adjacent to $p$, and in the maximal case, on the convexity of the position of the connected components of $A\setminus T_p A$. We study the combinatorial properties and decompositions of dessins corresponding to real uninodal trigonal curves in real ruled surfaces. Uninodal dessins in any surface with non-empty boundary 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 generic pointed quartic curves in $\mathbb{R}\mathbb{P}^{2}$. This classification was first obtained by S. Rieken based on the relation between quartic curves and del Pezzo surfaces.