Triple covers of algebraic surfaces and a generalization of Zariski's example

Abstract

Let $B$ be a reduced sextic curve in $\mathbb{P}^2$. In the case when singularities of $B$ are only six cusps, Zariski proved that there exists a non-Galois triple cover branched at $B$ if and only if $B$ is given by an equation of the form $G_2^3 + G_3^2$, where $G_i$ denotes a homogeneous polynomial of degree $i$. In this article, we generalize Zariski's statement to any reduced sextic curve with at worst simple singularities. To this purpose, we give formulae for numerical invariants of non-Galois triple covers by using Tan's canonical resolution.