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The multiplicity of a Mori Dream Space

Michele Rossi

TL;DR

This work generalizes the multiplicity concept from fake weighted projective spaces to Mori Dream Spaces by leveraging canonical toric embeddings and the associated weight data. It introduces the weight group $G_Q$ and weight modulus $|Q|$, establishes divisibility of multiplicity by the weight order $g_Q$, and reveals a rich interplay between polar duality, degree, and topological data via étale fundamental groups. A central achievement is a topological classification of Fano toric varieties and their polar partners, extended to $ ext{Q}$-Gorenstein and $ ext{Q}$-Fano cases, using subgroups of $G_Q$ and their complements. The paper then extends bounds on multiplicity to $ ext{Q}$-Gorenstein toric varieties and to Mori Dream Spaces, showing that multiplicity can be controlled through the ambient toric data, unitary 1-coverings, and the movable cone; it provides concrete bounds and a roadmap for classifying these varieties by weight data and Gorenstein index. Overall, the results illuminate how algebraic, geometric, and topological structures cohere in toric ambient spaces to bound and classify complex birational geometries in the Mori framework.

Abstract

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.

The multiplicity of a Mori Dream Space

TL;DR

This work generalizes the multiplicity concept from fake weighted projective spaces to Mori Dream Spaces by leveraging canonical toric embeddings and the associated weight data. It introduces the weight group and weight modulus , establishes divisibility of multiplicity by the weight order , and reveals a rich interplay between polar duality, degree, and topological data via étale fundamental groups. A central achievement is a topological classification of Fano toric varieties and their polar partners, extended to -Gorenstein and -Fano cases, using subgroups of and their complements. The paper then extends bounds on multiplicity to -Gorenstein toric varieties and to Mori Dream Spaces, showing that multiplicity can be controlled through the ambient toric data, unitary 1-coverings, and the movable cone; it provides concrete bounds and a roadmap for classifying these varieties by weight data and Gorenstein index. Overall, the results illuminate how algebraic, geometric, and topological structures cohere in toric ambient spaces to bound and classify complex birational geometries in the Mori framework.

Abstract

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 and the weight modulus of a complete toric variety. Their topological interpretation provides a framework for classifying Fano and -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.

Paper Structure

This paper contains 34 sections, 41 theorems, 228 equations, 2 figures.

Table of Contents

  1. Multiplicity beyond Fake Weighted Projective Spaces
  2. The Weight Group and its relation with Multiplicity
  3. Topological significance of the Weight Group
  4. The Weight Modulus and the Degree of a Fano toric variety
  5. Preliminaries and notation
  6. Notation on toric varieties
  7. List of notation
  8. Notation on matrices
  9. Matrices, cones and polytopes
  10. $F$-matrices, $W$-matrices and $CF$-matrices
  11. The GKZ-decomposition: cells and chambers
  12. Isomorphisms in codimension 1 and small $\mathbb{Q}$-factorial modification
  13. Bounds on Fano Toric Varieties
  14. Fano-Gorenstein condition on the weight matrix
  15. Multiplicity of a complete toric variety
  16. ...and 19 more sections

Key Result

Theorem 1

Let $X$ be a $n$-dimensional fake wps with at worst canonical singularities. In each case, equality is achieved by a unique $X$.

Figures (2)

  • Figure 1: Ex. \ref{['ex:blupP3']}: the section of the cone $\mathop{\mathrm{Mov}}\nolimits(V)$ and its chambers, inside the pseudo-effective cone $\overline{\mathop{\mathrm{Eff}}\nolimits}(V)$, as cut out by the plane $\sum_{i=1}^3x_i=1$.
  • Figure 2: Example \ref{['ex:QFanoCanonica']}: the polytope $\mathop{\mathrm{Conv}}\nolimits(\Lambda)$ spanning the fan $\Xi^\circ$.

Theorems & Definitions (100)

  • Theorem 1: Thm. 1.1. in AKLN
  • Theorem 2: Prop. 5.5 in Conrads
  • Definition 1.1: Fan and weight matrices
  • Definition 1.2: Bunch of cones associated with a fan
  • Remark 1.3
  • Definition 1.4: Def. 15 in R-wMDS
  • Proposition 1.5
  • Theorem 2.1
  • proof
  • Definition 2.2
  • ...and 90 more