Normality of the higher product-orbifold semigroups

Determine whether the affine semigroup associated with the product orbifold $(S^1/Z_2)^k$ is normal for every $k\ge3$.

Background

For every k≥3k\ge3, the paper proves that the semigroup of (S1/Z2)k(S^1/Z_2)^k is not a complete intersection and is not a cut ideal. The cases k=1k=1 and k=2k=2 are known to be normal.

The unresolved issue is whether normality persists for all higher products. A positive answer would, by Hochster’s theorem, produce an infinite family of normal, Cohen–Macaulay, non-complete-intersection toric rings.

References

Whether $S_k$ is normal for $k\ge3$ is not settled here.

— 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, “Every $k\ge3$, in one line,” paragraph “The question this leaves open”