Weierstrass pure gaps on curves with three distinguished points

Abstract

Let $\mathbb{K}$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $\mathbb{K}$, containing three points, $P_1,P_2,$ and $P_3$, such that $n{P}_1+P_2$, $n{P}_2+P_3$, and $n{P}_3+P_1$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_1,P_2,P_3\}$, as well as an explicit description of all pure gaps at any subset of $\{P_1,P_2,P_3\}$. When $\mathbb{K}={\overline{\mathbb{F}}}_q$, this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound.

12 pages