Isomorphism of the three spherical Hilbert-scheme models

Determine whether the canonical birational morphisms between Kivinen’s symbolic, diagonal, and sign Hilbert-scheme models associated with a finite real reflection group are isomorphisms in general.

Background

For a finite real reflection group, the paper discusses three models constructed from ordinary and symbolic powers of the reflection arrangement ideals: the sign model, the diagonal model, and the symbolic model. These models are related by canonical birational morphisms.

The paper proves that the three spherical models coincide asymptotically for type B_n/C_n, but it does not settle whether the corresponding birational morphisms are isomorphisms for arbitrary reflection groups or outside the stated asymptotic type-B setting.