On the Weierstrass semigroups of n points of a smooth curve: an addendum

Abstract

Let X be a smooth curve of genus g > 0. For any ${n\geq 2}$ and any n distinct points ${P_1,\dots ,P_n\in X}$, let ${H(P_1,\dots ,P_n)}$ be the set of all ${(a_1,\dots ,a_n)\in \mathbb {N}^n}$ such that ${\mathcal {O}_X(a_1P_1+\cdots +a_nP_n)}$ is spanned. Let e(g, n) be the maximum of all ${a_1+\cdots +a_n}$ among all ${(a_1,\dots ,a_n)}$ in a minimal subset of ${H(P_1,\dots ,P_n)}$ generating it as a semigroup, for some ${P_1,\dots ,P_n}$ and some X of genus g. We prove that ${e(g,n) \leq 3g-1}$ and that e(g, n) = 3g − 1 in characteristic 0.