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

In this article we obtain a rigid isotopy classification of generic pointed quartic curves (A, p) in $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 by studying the combinatorial properties of dessins. The dessins are real versions, proposed by Orevkov (Ann Fac Sci Toulouse 12(4):517–531, 2003), 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 $$T_p A$$ T p A intersects, the nature of connected components of $$A\setminus T_p A$$ A \ T p A adjacent to p, and in the maximal case, on the convexity of the position of the connected components of $$A\setminus T_p A$$ A \ 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 $${\mathbf {D}}^2$$ 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 $${\mathbb {R}}{\mathbb {P}}^{2}$$ R P 2 . This classification was first obtained in Rieken (Geometr Ded 185(1):171–203, 2016) based on the relation between quartic curves and del Pezzo surfaces.

Maximally inflected real trigonal curves on Hirzebruch surfaces

In 2014 A. Degtyarev, I. Itenberg, and the author gave a description, up to fiberwise equivariant deformations, of maximally inflected real trigonal curves of type I (over a base B B of an arbitrary genus) in terms of the combinatorics of sufficiently simple graphs and for B = P 1 B=\mathbb {P}^1 obtained a complete classification of such curves. In this paper, the mentioned results are extended to maximally inflected real trigonal curves of type II over B = P 1 B=\mathbb {P}^1 .

Counting the number of trigonal curves of genus 5 over finite fields

The trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.