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Cluster type varieties

Joaquín Moraga

Abstract

Cluster type varieties are compactifications of algebraic tori on which the volume form has no zeros. These form a natural class of varieties that generalizes both toric varieties and cluster varieties. The aim of this article is to introduce the reader to the concept of cluster type varieties and explain some recent results towards the understanding of these varieties. At the same time, we will pose some problems for further research.

Cluster type varieties

Abstract

Cluster type varieties are compactifications of algebraic tori on which the volume form has no zeros. These form a natural class of varieties that generalizes both toric varieties and cluster varieties. The aim of this article is to introduce the reader to the concept of cluster type varieties and explain some recent results towards the understanding of these varieties. At the same time, we will pose some problems for further research.
Paper Structure (10 sections, 33 theorems, 31 equations)

This paper contains 10 sections, 33 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Toric geometry
  3. Cluster geometry
  4. Singularity theory
  5. Birational geometry
  6. The cluster type condition in dimension three
  7. Complexity, birational complexity, and the cluster type condition
  8. Finite actions on Fano varieties
  9. The cluster type condition in families
  10. Cox rings of cluster type varieties

Key Result

Theorem 2.4

Let $X$ be a $n$-dimensional projective variety. The variety $X$ is toric if and only if ${\rm Aut}^0(X)$ has reductive rank $n$.

Theorems & Definitions (73)

  • Definition 2.1
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 63 more