Part IX-B · An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them (Part IX-B)

The equivalence classes of orbifold boundary conditions are the fibres of a marginal map, and are indexed by the elements of an affine semigroup. This note identifies that semigroup.

When the alphabet carries no weights the object already has a name and a literature, and the note says so plainly: over Z₂ it is the cut configuration of an explicit graph in the sense of Sturmfels and Sullivant — the four-cycle for T²/Z₂ and the wheel W₄ for S¹/Z₂ × S¹/Z₂, verified as an equality of point configurations; over Z_m with equal cone orders it is a group-based phylogenetic model on a claw tree; and for the higher products it is the binary hierarchical model of the boundary complex of a cross-polytope. The ring of the product orbifold is a row of a 2008 table — codimension, degree, minimal generators, normality — every invariant of which this machinery reproduced without knowing that table existed.

What is established here: thirteen gluing trees, the unitary, orthogonal and symplectic realisations of every case; the group-based model on the tripod for every finite abelian group — a complete intersection exactly when |G| ≤ 3; that (S¹/Z₂)^k is not a complete intersection for any k ≥ 3 and is not a cut ideal either, for a reason that is a count of edges; the exceptional case as a list of invariants rather than a negation — normal, Cohen–Macaulay, normalised volume 64, not a complete intersection, Cohen–Macaulay type 9; and that its minimal Markov basis is unique up to signs, sixteen indispensable binomials in exactly two orbits of eight under an automorphism group of order 128.

One distinction is drawn explicitly because the vocabulary collides: Casanellas, Fernández-Sánchez and Michałek prove these varieties are complete intersections in a Zariski open set; the question here is the global one, about the toric ideal itself, and the Z₄ tripod separates the two.

Companion to Part IX-A: that note determines which generators exist, this one which relations hold among them. Paper in English and Spanish; every ancillary script and its receipt travel with the record.