A note on normal triple covers over P² with branch divisors of degree 6

Abstract

Let S and T be reduced divisors on P² which have no common components, and Δ = S + 2T. We assume deg Δ = 6. Let π : X → P² 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 P¹-bundle over an elliptic curve or a normal cubic surface in P³. Consequently, we give a necessary and sufficient condition for Δ to be the branch divisor of a normal triple cover over P².