Criteria for proving affine semigroups are not complete intersections

Develop effective criteria for establishing that an affine semigroup is not a complete intersection.

Background

The paper compares the product-orbifold semigroups with examples that have the same numbers of extreme rays, dimension, rank, and a similar sign pattern, yet differ in whether they are complete intersections. This shows that coarse numerical data do not determine the property.

The cited state-of-the-art observation identifies the broader unresolved problem of finding structural criteria—potentially involving support, circuits, or oriented matroids—that certify failure of the complete-intersection property.

References

finding good criteria for establishing that $S$ is not a complete intersection remains an interesting open problem.

— An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them  (2609.02630 - Marín, 2 Sep 2026) in Section 9, paragraph “The exact ties”