Generating systems, generalized Thomsen collections and derived categories of toric varieties
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912755276054528 |
|---|---|
| author | Yi, Xiaodong |
| author_facet | Yi, Xiaodong |
| contents | Bondal claims that for a smooth toric variety $X$, its bounded derived category of coherent sheaves $D_{c}^{b}(X)$ is generated by the Thomsen collection $T(X)$ of line bundles obtained as direct summands of the pushforward of $\mathcal{O}_{X}$ along a Frobenius map with sufficiently divisible degree. The claim is confirmed recently. In this article, we consider a generalized Thomsen collection of line bundles $T(X,D)$ with a $\mathbb{Q}$-divisor $D$ as an auxiliary input, which recovers Thomsen's oringinal collection by setting $D=0$. We introduce the notion of a generating system and prove a theorem on the generation of $\mathcal{O}_{X}$ using many line bundles arising from the generating system. As an application, we verify Bondal's claim for some toric varieties, using a different argument from existing works. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generating systems, generalized Thomsen collections and derived categories of toric varieties Yi, Xiaodong Algebraic Geometry Bondal claims that for a smooth toric variety $X$, its bounded derived category of coherent sheaves $D_{c}^{b}(X)$ is generated by the Thomsen collection $T(X)$ of line bundles obtained as direct summands of the pushforward of $\mathcal{O}_{X}$ along a Frobenius map with sufficiently divisible degree. The claim is confirmed recently. In this article, we consider a generalized Thomsen collection of line bundles $T(X,D)$ with a $\mathbb{Q}$-divisor $D$ as an auxiliary input, which recovers Thomsen's oringinal collection by setting $D=0$. We introduce the notion of a generating system and prove a theorem on the generation of $\mathcal{O}_{X}$ using many line bundles arising from the generating system. As an application, we verify Bondal's claim for some toric varieties, using a different argument from existing works. |
| title | Generating systems, generalized Thomsen collections and derived categories of toric varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2506.23531 |