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