Weierstrass pure gaps on curves with three distinguished points

TL;DR

This paper studies pure gaps on specific smooth plane curves with three distinguished points, providing explicit descriptions and dimension results, with applications to constructing better algebraic geometry codes.

Contribution

It explicitly describes pure gaps and divisor dimensions on a class of curves with three special points, extending understanding of their algebraic and coding-theoretic properties.

Findings

Explicit description of pure gaps at subsets of three points.

Determination of divisor dimensions supported on three points.

Construction of algebraic geometry codes with improved minimum distance.

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 $nP_1+P_2$, $nP_2+P_3$, and $nP_3+P_1$ 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.