三点曲线上的Weierstrass纯间隙
Weierstrass pure gaps on curves with three distinguished points
摘要：
设$\mathbb{K}$是一个代数闭域.本文考虑$\mathbb{K}$上的次数为$n+1>3$的光滑平面曲线类，它包含三个点，$P_1,P_2,$和$P_3$，使得$nP_1+P_2$，$nP_2+P_3$和$nP_3+P_1$是由三条不同的线切割出来的因子.对于这样的曲线，我们确定了某些特殊因子支持$\{P_1,P_2,P_3\}$的尺寸，以及明确的描述在任何子集的$\{P_1,P_2,P_3\}$的所有纯间隙。当$\mathbb{K}=\overline{\mathbb{F}}_q$，这类曲线，其中包括埃尔米特曲线，被用来构造代数几何码具有最小距离优于Goppa界。
英文摘要：
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.
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