The uniqueness of Weierstrass points with semigroup \(\langle a;b\rangle \) and related semigroups

Assume a and \(b=na+r\) with \(n \ge 1\) 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 \(r \ne a-1\) and \(b \ne a+1\) we prove C has no other point \(Q \ne P\) 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.