Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2011.08288 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910021210603520 |
|---|---|
| author | Opper, Sebastian |
| author_facet | Opper, Sebastian |
| contents | This paper studies the class of spherical objects over any Kodaira $n$-cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the $n$-punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk and extends earlier results by Burban-Kreussler and Lekili-Polishchuk. The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla. The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk-Drozd-Greuel which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product we obtain a closed formula for the cyclic sequence of any simple vector bundle on $C_n$ as introduced by Burban-Drozd-Greuel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_08288 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras Opper, Sebastian Algebraic Geometry Representation Theory 16E35 (Primary) 14H45 (Secondary) This paper studies the class of spherical objects over any Kodaira $n$-cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the $n$-punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk and extends earlier results by Burban-Kreussler and Lekili-Polishchuk. The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla. The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk-Drozd-Greuel which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product we obtain a closed formula for the cyclic sequence of any simple vector bundle on $C_n$ as introduced by Burban-Drozd-Greuel. |
| title | Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras |
| topic | Algebraic Geometry Representation Theory 16E35 (Primary) 14H45 (Secondary) |
| url | https://arxiv.org/abs/2011.08288 |