Brieskorn-Pham singularities via ACM bundles on Geigle-Lenzing projective spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912205281165312 |
|---|---|
| author | Chen, Jianmin Ruan, Shiquan Weng, Weikang |
| author_facet | Chen, Jianmin Ruan, Shiquan Weng, Weikang |
| contents | We study the singularity category of the Brieskorn-Pham singularity $R=k[X_1, \dots, X_4]/(\sum_{i=1}^{4} X_i^{p_i})$, associated with the Geigle-Lenzing projective space $\mathbb{X}$ of weight quadruple $(p_1,\dots, p_4)$, by investigating the stable category $\underline{\mathsf{ACM}} \, \mathbb{X}$ of arithmetically Cohen-Macaulay bundles on $\mathbb{X}$.
We introduce the notion of $2$-extension bundles on $\mathbb{X}$, which is a higher dimensional analog of extension bundles on a weighted projective line of Geigle-Lenzing, and then establish a correspondence between $2$-extension bundles and a certain important class of Cohen-Macaulay $R$-modules studied by Herschend-Iyama-Minamoto-Oppermann. Furthermore, we construct a tilting object in $\underline{\mathsf{ACM}} \, \mathbb{X}$ consisting of $2$-extension bundles, whose endomorphism algebra is a $4$-fold tensor product of certain Nakayama algebras. We also investigate the Picard group action on $2$-extension bundles and obtain an explicit formula for the orbit number, which gives a positive answer to a higher version of an open question raised by Kussin-Lenzing-Meltzer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Brieskorn-Pham singularities via ACM bundles on Geigle-Lenzing projective spaces Chen, Jianmin Ruan, Shiquan Weng, Weikang Algebraic Geometry Representation Theory We study the singularity category of the Brieskorn-Pham singularity $R=k[X_1, \dots, X_4]/(\sum_{i=1}^{4} X_i^{p_i})$, associated with the Geigle-Lenzing projective space $\mathbb{X}$ of weight quadruple $(p_1,\dots, p_4)$, by investigating the stable category $\underline{\mathsf{ACM}} \, \mathbb{X}$ of arithmetically Cohen-Macaulay bundles on $\mathbb{X}$. We introduce the notion of $2$-extension bundles on $\mathbb{X}$, which is a higher dimensional analog of extension bundles on a weighted projective line of Geigle-Lenzing, and then establish a correspondence between $2$-extension bundles and a certain important class of Cohen-Macaulay $R$-modules studied by Herschend-Iyama-Minamoto-Oppermann. Furthermore, we construct a tilting object in $\underline{\mathsf{ACM}} \, \mathbb{X}$ consisting of $2$-extension bundles, whose endomorphism algebra is a $4$-fold tensor product of certain Nakayama algebras. We also investigate the Picard group action on $2$-extension bundles and obtain an explicit formula for the orbit number, which gives a positive answer to a higher version of an open question raised by Kussin-Lenzing-Meltzer. |
| title | Brieskorn-Pham singularities via ACM bundles on Geigle-Lenzing projective spaces |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2501.15375 |