The Woo-Yong conjecture

Abstract

We prove the Woo-Yong conjecture: the non-Gorenstein locus of a type A Schubert variety is the union of its generically non-Gorenstein maximal singular components. More strongly, at every scheme point, local Weil divisor classes are detected at the generic points of the maximal singular components containing that point. For arbitrary finite crystallographic type, the analogous localization map is injective on the torsion subgroup of the source local class group. This determines Cartier indices of locally $\mathbb{Q}$-Cartier divisors and gives the Woo-Yong criterion on the $\mathbb{Q}$-Gorenstein locus. A type D family shows the necessity of this hypothesis: the ambient variety is generically Gorenstein along its singular locus but not $\mathbb{Q}$-Gorenstein along a deeper nonempty locus. The type A proof and the counterexamples are geometric; the general torsion proof also uses bounded finite inputs with explicit certificates and reproducible reconstruction programs.