The multiplicity of a Mori Dream Space

In this paper we extend the concept of multiplicity from fake weighted projective spaces, as considered by Averkov, Kasprzyk, Lehmann and Nill in 2021, to Mori Dream Spaces, exploring interesting connections between the algebraic, geometric, and topological properties of these varieties. To this end, we introduce the weight group $G_Q$ and the weight modulus $|Q|$ of a complete toric variety. Their topological interpretation provides a framework for classifying Fano and $\mathbb{Q}$-Fano toric varieties, offering an alternative approach for a further understanding of this rich and fascinating area of algebraic geometry. In particular, we exhibit an algebraic interpretation of Batyrev's polar duality between Fano toric varieties as a direct sum decomposition of their common weight group.