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

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.