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

Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common components, and $Δ=S+2\,T.$ We assume $\degΔ=6.$ Let $π:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $Δ,$ i.e. $π$ 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 $Δ$ to be the branch divisor of a normal triple cover over $\mathbb{P}^2.$

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