Fano compactifications of mutation algebras
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917169784160256 |
|---|---|
| author | Enwright, Joshua Francone, Luca Moraga, Joaquín Spink, Hunter |
| author_facet | Enwright, Joshua Francone, Luca Moraga, Joaquín Spink, Hunter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fano compactifications of mutation algebras Enwright, Joshua Francone, Luca Moraga, Joaquín Spink, Hunter Algebraic Geometry Combinatorics Representation Theory Primary 14M25, 14E25, Secondary 14B05, 14E30, 14E05 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. |
| title | Fano compactifications of mutation algebras |
| topic | Algebraic Geometry Combinatorics Representation Theory Primary 14M25, 14E25, Secondary 14B05, 14E30, 14E05 |
| url | https://arxiv.org/abs/2512.21839 |