Weierstrass pure gaps on curves with three distinguished points

2021/07/17 by Borges, Herivelto, Duran, Gregory · 1 citation
#14G50 #14H55 #94B05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2107.08290

Abstract

Let \mathbbK be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree n+1>3 over \mathbbK, containing three points, P₁,P₂, and P₃, such that nP₁+P₂, nP₂+P₃, and nP₃+P₁ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on \P₁,P₂,P₃\, as well as an explicit description of all pure gaps at any subset of \P₁,P₂,P₃\. When \mathbbK=\mathbbF_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.

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