The multiplicity of a Mori Dream Space: applications to polar duality and the classification of toric $\mathbb{Q}$-Fano varieties
Year: 2025
Type: preprint
Abstract: We extend the multiplicity of fake weighted projective spaces, as introduced by Averkov, Kasprzyk, Lehmann and Nill in 2021, to complete toric varieties and Mori Dream Spaces, defining it as the order of the torsion subgroup of the divisor class group. It equals the degree of the canonical $1$-covering, a finite surjective morphism étale in codimension one. We classify toric $\Q$-Fano varieties with prescribed weight matrix and Gorenstein index by admissible subgroups of a finite abelian group, realized as the deck transformation groups of their universal $1$-coverings. The classification allows arbitrary class-group rank without assuming $\Q$-factoriality, and yields an intrinsic intermediate $1$-covering, the \emph{unitary} one, controlling the Gorenstein index. We outline a computational classification strategy based on weight matrices and finite subgroup enumeration. In the Gorenstein Fano case, a perfect pairing of weight groups expresses polar duality as an inclusion-reversing subgroup correspondence. For iso-weighted polar partners over $\C$, the mirror correspondence admits an internal description within the anticanonical family of their common universal $1$-covering, through invariant subfamilies and their quotient actions. Finally, we derive divisibility relations and multiplicity bounds, also applicable to Mori Dream Spaces whose canonical toric ambient varieties admit suitable $\Q$-Fano completions.
Source: arXiv (Cornell University)
Author: Rossi, Michele
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407764311