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

Abstract

Let $S$ and $T$ be reduced divisors on ${\mathbb{P}}^2$ which have no common components, and $\Delta =S+2T.$ We assume $\mathrm{deg}\Delta =6.$ 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 ${\mathbb{P}}^2.$

9 pages