A note on normal triple covers over \(\mathbb P^{2}\) with branch divisors of degree 6 (Q740758)

In spite of several papers on triple covers, the basic problem of characterizing branch divisors of normal triple covers of the projective plane \(\mathbb P^2\) is still open. In the paper under review, the author deals with the case of a branch divisor of degree six, lower degree branch divisors being already studied in the literature. Let \(S\) and \(T\) be two reduced divisors on \(\mathbb P^2\), with no common components. The author aims at describing normal triple covers \(\pi:X \to \mathbb P^2\), with branch divisor \(\Delta = S+2T\) (i.e., with ramification index \(2\) over \(S\) and \(3\) over \(T\)). Let \(F \subset \mathbb P^2 \times \mathbb P^{2 \vee}\) be the flag variety of pairs of points and lines in \(\mathbb P^2\), and let \(p:F \to \mathbb P^2\) and \(q:F \to \mathbb P^{2 \vee}\) be the projections. Relying on the work on triple covers done by \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)], the author proves the following result. Let \(\pi:X \to \mathbb P^2\) be a normal triple cover whose branch locus \(\Delta\) has degree six: then either (i) \(\Delta =S\) is an irreducible sextic curve with nine cusps, \(X=q^{-1}(S^{\vee})\) (a \(\mathbb P^1\) bundle over the smooth plane cubic \(S^{\vee}\), dual to \(S\)), and \(\pi\) is the restriction of \(p\) to \(X\), or (ii) \(X\) is a cubic surface in \(\mathbb P^3\) and \(\pi\) is a projection from a point of \(\mathbb P^3 \setminus X\). Furthermore (i) occurs if and only if the nine cusps are total branching points of \(\pi\). According to a previous result of \textit{H. Ishida} and \textit{H.-o Tokunaga} [Adv. Stud. Pure Math. 56, 169--185 (2009; Zbl 1198.14016)], if the branch divisor is a sextic curve with simple singularities, then either \(X\) is a quotient of an abelian surface by an involution, or \(X\) is as in (ii).