Admissible subcategories supported on curves
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911015022624768 |
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| author | Pirozhkov, Dmitrii |
| author_facet | Pirozhkov, Dmitrii |
| contents | Let $X$ be a smooth projective variety. We study admissible subcategories of the bounded derived category of coherent sheaves on $X$ whose support is a proper subvariety $Z \subset X$. We show that any one-dimensional irreducible component of $Z$ is a rational curve. When $\operatorname{dim} Z = 1$, we prove that at least one irreducible component in $Z$ intersects the canonical class $K_X$ negatively. In particular, this implies that a surface with a nef and effective canonical bundle has indecomposable derived category, confirming the conjecture by Okawa. We also prove that a configuration of curves with non-negative self-intersections on a surface cannot support an admissible subcategory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16567 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Admissible subcategories supported on curves Pirozhkov, Dmitrii Algebraic Geometry Let $X$ be a smooth projective variety. We study admissible subcategories of the bounded derived category of coherent sheaves on $X$ whose support is a proper subvariety $Z \subset X$. We show that any one-dimensional irreducible component of $Z$ is a rational curve. When $\operatorname{dim} Z = 1$, we prove that at least one irreducible component in $Z$ intersects the canonical class $K_X$ negatively. In particular, this implies that a surface with a nef and effective canonical bundle has indecomposable derived category, confirming the conjecture by Okawa. We also prove that a configuration of curves with non-negative self-intersections on a surface cannot support an admissible subcategory. |
| title | Admissible subcategories supported on curves |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2506.16567 |