Fano compactifications of mutation algebras

TL;DR

This work introduces mutation semigroup algebras (MSA) as a unifying framework for cluster algebras and semigroup algebras, linking algebraic structure to birational and Fano geometry. It establishes that spectra of MSAs with klt singularities admit log Fano compactifications, and, when $U$ is $\mathbb{Q}$-factorial, these compactifications can be chosen $\mathbb{Q}$-factorial; it also proves a characterization of cluster type Fano varieties via their Cox rings as graded MSAs. The paper provides extensive explicit examples from Lie theory and birational geometry, showing many open Schubert, Richardson, and brick varieties are cluster type and log Fano, and it develops two Cox-ring constructions for blow-ups and cubic surfaces. Overall, the results bridge cluster-type geometry, Fano theory, and Cox-ring structures, offering a coherent framework for studying algebras arising from mutations in diverse contexts.

Abstract

In this article, we introduce the notion of mutation semigroup algebras. This concept simultaneously generalizes cluster algebras and semigroup algebras. We show that, under some mild conditions on the singularities, the spectrum $U={\rm Spec}(R)$ of a mutation semigroup algebra $R$ admits a log Fano compactification $U\hookrightarrow X$. The compactification $X$ can be chosen to be a $\mathbb{Q}$-factorial log Fano variety whenever $U$ is $\mathbb{Q}$-factorial. Furthermore, we prove that a $\mathbb{Q}$-factorial klt Fano variety $X$ is of cluster type if and only if its Cox ring ${\rm Cox}(X)$ is a ${\rm Cl}(X)$-graded mutation semigroup algebra. In order to enlighten the previous theorems, we provide several explicit examples motivated by birational geometry, representation theory, and combinatorics.

Theorems & Definitions (77)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Definition 2.1
• Theorem 2.2