The multiplicity of a Mori Dream Space: applications to polar duality and the classification of toric $\mathbb{Q}$-Fano varieties

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 $1$-cover