I‑AIMS Institute for Artificial Intelligence
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Hulls, linear equivalence, and weighted superelliptic codes

J. Mezinaj, Tanush Shaska

I-AIMS preprint sh-140 · 2026

The containment of the code of the meet G ∧ A in the hull, and the identity G ∨ A − D = K − G ∧ A exchanging meet and join, are known, and imposing that G ∧ A be principal is how algebraic geometry codes with one-dimensional hull are built. We turn that construction into a measurement. For arbitrary divisors G and A we compute C_L(D, G) ∩ C_L(D, A) exactly: it is the code of the meet together with an excess ε(G, A), canonically the quotient of the two. In the dual situation A = D − G + K the excess is a subquotient of H¹(X, 𝒪_X(G ∧ A))^∨, so it vanishes when the meet is non-special, and where it does not vanish K − G ∧ A is linearly equivalent to an effective divisor. On superelliptic curves yⁿ = f(x), whose weighted plane models in ℙ²_((1, n/c, d/c)), c = gcd(n, d), turn every Riemann-Roch dimension into a lattice count, this divisor-class condition holds automatically whenever its degree is nonnegative, and the excess detects instead the finer condition that a section be decomposable along G and A. The meet and the join are invariant under s ↦ M − s, so every asymmetry of the hull profile is excess: on a totally split genus-two curve over 𝔽₇ the reflected degrees s = 1 and s = 4 present the same class and have different excess. On the range where the excess is blind the hull is c μ(s) − nδ + 1 − g_X, μ(s) = min{s, M − s}, and depends on the curve only through its number of affine rational points. Over a prime field, under one explicit inequality on (n, d, q), ℓ(⌊ M/2 ⌋ D_∞) bounds the hull dimension throughout the family, with equality exactly on the totally split locus.

MSC classes: Primary 94B27, 14G50; Secondary 14C20, 14H51, 14M25, 11T71

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