Weierstrass Semigroups Satisfying MP Equalities and Curves on Toric Surfaces

A numerical semigroup H is said to be cyclic if it is the Weierstrass semigroup of a total ramification point of some cyclic covering of the projective line. In this case, the elements of the standard basis of H satisfy numerical conditions that we have chosen to term MP equalities, after Morrison and Pinkham who first proved them. The converse is not true, as there are semigroups satisfying the MP equalities that are not cyclic. In this paper, we consider the situation of a smooth curve C lying on a smooth compact toric surface S acted on by the torus T which is a dense open subset of S. We prove that the Weierstrass semigroup of a total ramification point of the restriction to C of a toric fibration of S, which lies on some T-invariant divisor, is cyclic if and only if it satisfies the MP equalities.