2026/08/03 by Jurgen Mezinaj, Tanush Shaska
Mathematics · Computer Science · #math.AG #cs.IT #math.IT #msc:94B27 #msc:14G50 #msc:14C20 #msc:14H51 #msc:14M25 #msc:11T71
39 pages, 6 tables
arxiv created 2026/08/03 · arxiv updated 2026/08/05
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 CL(D,G)∩ CL(D,A) exactly: it is the code of the meet together with an excess ε(G,A), canonically their quotient and a subquotient of H1(\mathcal O(G\wedge A)). So ε 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 yn=f(x) both sides can be computed: their weighted plane models in \mathbb P2(1,n/c,d/c), c=gcd(n,d), identify codes Cs of weighted forms of degree s with those of sD_∞ and turn hulls into lattice counts. The range on which ε is blind is an explicit interval of degrees, where \dimHull(Cs)=cμ(s)-nδ+1-gX, μ(s)=min\s,M-s\, depends only on its affine-point count. Outside it the meet and join are invariant under s↦ M-s while ε 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 F7 and over \mathbb F11, present the same class at the same pair of degrees and are separated by the profile alone. If 0≤°(G\wedge A)≤2gX-2 the hull is at most gX+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 ℓ(\lfloor M/2\rfloor D_∞), attained exactly on the totally split locus.