Higher rank Clifford's theorem on the smooth quadric
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918163254345728 |
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| author | Raha, Neelarnab |
| author_facet | Raha, Neelarnab |
| contents | Brill-Noether theory of curves has played a crucial role in the study of curves and their moduli since the 19th century, and has been extensively studied by several authors. Clifford's theorem provides a starting point in determining the emptiness of Brill-Noether loci by providing an upper bound on $h^0(L)$ for a line bundle $L$ on a smooth curve $C$ in terms of the degree of $L$. It also characterizes the cases for which equality holds.
In this paper, we prove an analogous result for higher rank sheaves on $\mathbb{P}^1\times\mathbb{P}^1$. Depending on how nice the first Chern class is, and whether the sheaf has global generation properties, we prove sharp upper bounds on $h^0(E)$ for slope semistable sheaves $E$ in terms of $\operatorname{rk}(E)$ and $c_1(E)$. We also find that any $E$ achieving the bound is a twist of a Steiner-like bundle, or closely related to such a bundle. As part of our investigation, we show that general extensions of stable vector bundles on elliptic curves and del Pezzo surfaces are semistable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher rank Clifford's theorem on the smooth quadric Raha, Neelarnab Algebraic Geometry Primary: 14J60, 14J26. Secondary: 14D20 Brill-Noether theory of curves has played a crucial role in the study of curves and their moduli since the 19th century, and has been extensively studied by several authors. Clifford's theorem provides a starting point in determining the emptiness of Brill-Noether loci by providing an upper bound on $h^0(L)$ for a line bundle $L$ on a smooth curve $C$ in terms of the degree of $L$. It also characterizes the cases for which equality holds. In this paper, we prove an analogous result for higher rank sheaves on $\mathbb{P}^1\times\mathbb{P}^1$. Depending on how nice the first Chern class is, and whether the sheaf has global generation properties, we prove sharp upper bounds on $h^0(E)$ for slope semistable sheaves $E$ in terms of $\operatorname{rk}(E)$ and $c_1(E)$. We also find that any $E$ achieving the bound is a twist of a Steiner-like bundle, or closely related to such a bundle. As part of our investigation, we show that general extensions of stable vector bundles on elliptic curves and del Pezzo surfaces are semistable. |
| title | Higher rank Clifford's theorem on the smooth quadric |
| topic | Algebraic Geometry Primary: 14J60, 14J26. Secondary: 14D20 |
| url | https://arxiv.org/abs/2510.16323 |