Non-Galois Triple Covering of $\mathbb{P}^2$ branched along quintic curves and their cubic equations

Abstract

Let $\varpi : S \to \mathbb{P}^2$ be a non-Galois triple covering given by the cubic equation $\zeta^3+3u\zeta+2v=0$, where $u$ and $v$ denote inhomogeneous coordinates of $\mathbb{P}^2$. Let $\hat{\pi} : \hat{X} \to \mathbb{P}^2$ be a $D_6$-covering of $\mathbb{P}^2$ branched along a quintic. There are two possibilities for the ramification types of $\hat{\pi}$. One is that $\hat{\pi}$ has the ramification index 2 (resp. 3) along a conic (resp. a cubic), and the other is that $\hat{\pi}$ has the ramification index 2 (resp. 3) along a quartic (resp. a line). There exist 18 types in the latter case ([8]). For each $\hat{\pi}$ of the 18 types, there exists a non-Galois triple covering $\pi : X \to \mathbb{P}^2$ with the same branch locus as $\hat{\pi}$. In this article, we study rational maps $\Phi : \mathbb{P}^2 \to \mathbb{P}^2$ such that the pull-backs of $\varpi$ by $\Phi$ give rise to $\pi : X \to \mathbb{P}^2$.