Horospherical varieties with quotient singularities

Our main result is a combinatorial characterization of when a horospherical variety has (at worst) quotient singularities. Using this characterization, we show that every quasiprojective horospherical variety with quotient singularities is globally the quotient of a smooth variety by a finite abelian group.

💡 Research Summary

This paper provides a comprehensive study of quotient singularities on horospherical varieties. Quotient singularities, which are locally modeled on the quotient of a smooth variety by a finite group action, represent a fundamental class of mild singularities in algebraic geometry.

The first main result (Theorem 1.7) is a complete combinatorial characterization of when a horospherical variety X has (at worst) quotient singularities. The characterization is given in terms of the associated “coloured fan” Σ^c, a combinatorial datum that generalizes the fan in toric geometry. The theorem states that X has quotient singularities if and only if its coloured fan Σ^c is both simplicial and vivid.

• Simplicial means that for each coloured cone σ^c in Σ^c, the multiset consisting of the minimal generators of its non-coloured rays and the colour points from its colour set F is linearly independent. This condition is equivalent to X being Q-factorial.
• Vivid is a more subtle condition on the configuration of the “colours” (which correspond to certain B-invariant divisors) within the fan. It ensures that the local structure of the variety is, in a precise sense, constrained to be of a toric nature. This condition bridges the gap between being Q-factorial and having quotient singularities, analogous to how vividness (combined with regularity) bridges the gap between being factorial and being smooth for horospherical varieties.

The second main result (Theorem 1.2) answers a version of “Fulton’s Question” affirmatively for the class of quasiprojective horospherical varieties. It proves that every quasiprojective horospherical variety X with quotient singularities is globally the quotient of a smooth variety V by a finite abelian group F (i.e., X ≅ V / F). The proof leverages the combinatorial characterization from Theorem 1.7. It first shows that for such an X, the “universal Cox variety” obtained via the Cox construction (a standard tool for spherical varieties) is smooth. A known result from