Hulls, linear equivalence, and weighted superelliptic codes

Abstract

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.