The Weierstrass Semigroup of an m-tuple of Collinear Points on a Hermitian Curve

We examine the structure of the Weierstrass semigroup of an m-tuple of points on a smooth, projective, absolutely irreducible curve X over a finite field \(\mathbb{F}\). A criteria is given for determining a minimal subset of semigroup elements which generate such a semigroup where 2 \( \leq m \leq |\mathbb{F}|\). For all 2 mq + 1, we determine the Weierstrass semigroup of any m-tuple of collinear \(\mathbb{F}_{q^2}\)-rational points on a Hermitian curve y q + y = x q + 1.