Cluster type varieties

TL;DR

This paper introduces cluster type varieties, a class of algebraic varieties generalizing toric and cluster varieties, discusses recent progress in understanding them, and suggests directions for future research.

Contribution

It provides an overview of cluster type varieties, highlighting recent results and open problems in their study.

Findings

Cluster type varieties generalize toric and cluster varieties.

Recent results advance understanding of their structure.

Open problems suggest future research directions.

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.