发表机构
Oakland University（奥克兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究围绕代数几何码的包络，推导了交与并的恒等式，计算了超椭圆曲线码的包络维度，利用包络轮廓区分了有限域上的完全分裂曲线，明确了包络的特性与范围。
AI 中文摘要
已知交 $G\wedge A$ 的码包含在包络中，且存在交换交与并的恒等式 $G\vee A-D=K-G\wedge A$；要求 $G\wedge A$ 为主元可构造具有一维包络的代数几何码。我们将该构造转化为一种度量。对任意除子 $G$ 和 $A$，我们精确计算 $C_L(D,G)\cap C_L(D,A)$：它是交的码加上一个过剩项 $\varepsilon(G,A)$，$\varepsilon$ 本质上是 $C_L(D,G)\cap C_L(D,A)$ 的商，且是 $H^1(\mathcal O(G\wedge A))$ 的子商。因此，当交非特殊时 $\varepsilon$ 恰好消失；否则它表明 $K-G\wedge A$ 线性等价于一个有效除子，在次数为零时对应皮卡群中单个类的消失：包络检测线性等价，而非由线性等价构造。对于超椭圆曲线 $y^n=f(x)$，两边均可计算：它们在加权射影空间 $\mathbb P^2_{(1,n/c,d/c)}$（其中 $c=\gcd(n,d)$）中的加权平面模型，将次数为 $s$ 的加权形式的码 $C_s$ 与 $sD_\infty$ 的码等同起来，并将包络转化为格计数。$\varepsilon$ 不敏感的范围是一个明确的次数区间，其中 $\dim\operatorname{Hull}(C_s)=c\mu(s)-n\delta+1-g_X$，$\mu(s)=\min\{s,M-s\}$，仅取决于其仿射点的数量。在该区间外，交和并不在 $s\mapsto M-s$ 下不变，而 $\varepsilon$ 则不然，因此包络轮廓的每个不对称性都是过剩项，且 $s$ 的阈值细化了除子类：两个亏格为2的完全分裂曲线，分别在 $\mathbb F_7$ 和 $\mathbb F_{11}$ 上，在同一对次数下呈现相同的类，却仅被轮廓区分。若 $0\leq\deg(G\wedge A)\leq2g_X-2$，则包络至多为 $g_X+1$，因此仅在其不敏感的区域才较大；在素域上，在关于 $(n,d,q)$ 的明确不等式下，其在该族上的最大值为 $\ell(\lfloor M/2\rfloor D_\infty)$，恰好出现在完全分裂轨迹上。
英文摘要
The containment of the code of the meet $G\wedge A$ in the hull and the identity $G\vee A-D=K-G\wedge A$ exchanging meet and join are known; imposing that $G\wedge A$ be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors $G$ and $A$ we compute $C_L(D,G)\cap C_L(D,A)$ exactly: it is the code of the meet together with an excess $\varepsilon(G,A)$, canonically their quotient and a subquotient of $H^1(\mathcal O(G\wedge A))$. So $\varepsilon$ vanishes exactly when the meet is non-special; otherwise it certifies that $K-G\wedge A$ is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves $y^n=f(x)$ both sides can be computed: their weighted plane models in $\mathbb P^2_{(1,n/c,d/c)}$, $c=\gcd(n,d)$, identify codes $C_s$ of weighted forms of degree $s$ with those of $sD_\infty$ and turn hulls into lattice counts. The range on which $\varepsilon$ is blind is an explicit interval of degrees, where $\dim\operatorname{Hull}(C_s)=cμ(s)-nδ+1-g_X$, $μ(s)=\min\{s,M-s\}$, depends only on its affine-point count. Outside it the meet and join are invariant under $s\mapsto M-s$ while $\varepsilon$ is not, so every asymmetry of the hull profile is excess and the threshold in $s$ refines the divisor class: two totally split curves of genus two, over $\mathbb F_7$ and over $\mathbb F_{11}$, present the same class at the same pair of degrees and are separated by the profile alone. If $0\leq°(G\wedge A)\leq2g_X-2$ the hull is at most $g_X+1$, so it is large only where it is blind, and over a prime field, under an explicit inequality on $(n,d,q)$, its maximum over the family is $\ell(\lfloor M/2\rfloor D_\infty)$, attained exactly on the totally split locus.
Comments39 pages, 6 tables
