Trigonal Morsifications on Hirzebruch Surfaces with an appendix by E. Shustin

TL;DR

This paper classifies rigid isotopy classes of certain trigonal curves on Hirzebruch surfaces with maximal double points, linking them to morsifications of specific real singularities and analyzing their combinatorial properties.

Contribution

It provides a classification of these curves and demonstrates how to realize their morsifications through Newton diagram modifications.

Findings

Classification of rigid isotopy classes achieved

Connection established between curves and singularity morsifications

Method developed for realizing morsifications via Newton diagrams

Abstract

In this paper we obtain a classification of rigid isotopy classes of totally reducible trigonal curves lying on a Hirzebruch surface $\Sigma_n$, and having a maximal number of non-degenerated double points. Such curves correspond to morsifications of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ (the union of three smooth real branches intersecting each other with multiplicity $n$). We obtain this classification by studying combinatorial properties of dessins. In the appendix, we prove that any morsification of a totally real semiquasihomogeneous singularity of weight $(3,3n)$ can be realized (up to isotopy) by the restriction of the equation to the Newton diagram and adding monomials under the Newton diagram.