A note on normal triple covers over $\mathbb{P}^2$ with branch divisors of degree 6

TL;DR

This paper classifies normal triple covers over the projective plane with branch divisors of degree 6, showing they are either certain $P^1$-bundles over elliptic curves or cubic surfaces, and provides conditions for such divisors.

Contribution

It characterizes the structure of normal triple covers over $P^2$ with degree 6 branch divisors and establishes necessary and sufficient conditions for their existence.

Findings

X is either a $P^1$-bundle over an elliptic curve or a cubic surface in $P^3$

Provides criteria for divisors to be branch divisors of such covers

Classifies all such normal triple covers with degree 6 branch divisors

Abstract

Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common components, and $\Delta=S+2\,T.$ We assume $\deg\Delta=6.$ Let $\pi:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $\Delta,$ i.e. $\pi$ is ramified along $S$ (resp. $T$) with the index 2 (resp. 3). In this note, we show that $X$ is either a $\mathbb{P}^1$-bundle over an elliptic curve or a normal cubic surface in $\mathbb{P}^3.$ Consequently, we give a necessary and sufficient condition for $\Delta$ to be the branch divisor of a normal triple cover over $\mathbb{P}^2.$