Weierstrass n-semigroups with even n and curves on toric surfaces

Abstract

First, we give some Weierstrass semigroups which cannot be attained by any smooth curve on a smooth compact toric surface. Next, for any integer l ≥ 2 we describe the Weierstrass semigroup of a total ramification point of a cyclic covering of the projective line with degree l using l − 1 non-negative integers. And finally, we will study smooth curves C lying on a smooth compact toric surface S acted by the torus T which is a dense open subset of S . For an even integer n ≤ 10 we characterize the Weierstrass semigroup of a total ramification point of a cyclic covering of degree n which is the restriction to C of a toric fibration of S such that the ramification point lies on some T -invariant divisor.