Fano compactifications of mutation algebras

TL;DR

This paper introduces mutation semigroup algebras, generalizes cluster and semigroup algebras, and explores their geometric properties, including log Fano compactifications and characterizations of cluster type varieties.

Contribution

It defines mutation semigroup algebras, establishes conditions for their spectra to admit log Fano compactifications, and characterizes cluster type Fano varieties via their Cox rings.

Findings

Spectrum admits log Fano compactification under mild singularity conditions.

Q-factorial klt Fano varieties are of cluster type iff their Cox rings are mutation semigroup algebras.

Provides explicit examples linking birational geometry, representation theory, and combinatorics.

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.