The uniqueness of Weierstrass points with semigroup ⟨a; b⟩ and related semigroups

Assume a and b = na + r with n >= 1 and 0 < r < a are relatively prime integers. In case C is a smooth curve and P is a point on C with Weierstrass semigroup equal to < a; b > then C is called a C-a;b-curve. In case r not equal a - 1 and b not equal a + 1 we prove C has no other point Q not equal P having Weierstrass semigroup equal to < a; b >, in which case we say that the Weierstrass semigroup < a; b > occurs at most once. The curve C-a;b has genus (a - 1)(b - 1)/2 and the result is generalized to genus g < (a - 1)(b - 1)/2. We obtain a lower bound on g (sharp in many cases) such that all Weierstrass semigroups of genus g containing < a; b > occur at most once.