Martens and Mumford theorems for higher rank Brill--Noether loci
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929635110944768 |
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| author | Nazarlou, Parviz Asefi Bajravani, Ali Hitching, George H. |
| author_facet | Nazarlou, Parviz Asefi Bajravani, Ali Hitching, George H. |
| contents | Generalizing the Martens theorem for line bundles over a curve $C$, we obtain upper bounds on the dimension of the Brill--Noether locus $B^k_{n, d}$ parametrizing stable bundles of rank $n \ge 2$ and degree $d$ over $C$ with at least $k$ independent sections. This proves a conjecture of the second author and generalizes bounds obtained by him in the rank two case. We give more refined results for some values of $d$, including a generalized Mumford theorem for $n \ge 2$ when $d \le g - 1$. The statements are obtained chiefly by analysis of the tangent spaces of $B^k_{n, d}$. As an application, we show that for $n \ge 5$ the locus $B^2_{n, n(g-1)}$ is irreducible and reduced for any $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Martens and Mumford theorems for higher rank Brill--Noether loci Nazarlou, Parviz Asefi Bajravani, Ali Hitching, George H. Algebraic Geometry 14H60, 14H51 Generalizing the Martens theorem for line bundles over a curve $C$, we obtain upper bounds on the dimension of the Brill--Noether locus $B^k_{n, d}$ parametrizing stable bundles of rank $n \ge 2$ and degree $d$ over $C$ with at least $k$ independent sections. This proves a conjecture of the second author and generalizes bounds obtained by him in the rank two case. We give more refined results for some values of $d$, including a generalized Mumford theorem for $n \ge 2$ when $d \le g - 1$. The statements are obtained chiefly by analysis of the tangent spaces of $B^k_{n, d}$. As an application, we show that for $n \ge 5$ the locus $B^2_{n, n(g-1)}$ is irreducible and reduced for any $C$. |
| title | Martens and Mumford theorems for higher rank Brill--Noether loci |
| topic | Algebraic Geometry 14H60, 14H51 |
| url | https://arxiv.org/abs/2412.12719 |