Brill-Noether theory on the projective plane for bundles with many sections
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916412067414016 |
|---|---|
| author | Coskun, Izzet Huizenga, Jack Raha, Neelarnab |
| author_facet | Coskun, Izzet Huizenga, Jack Raha, Neelarnab |
| contents | The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension.
Let $E$ be a semistable sheaf on the projective plane. In this paper, we give an upper bound for $h^0(E)$ in terms of the rank $r$ and the slope $μ$ of $E$. We show that the bound is achieved precisely when $E$ is a twist of a Steiner bundle. We classify the sheaves $E$ such that $h^0(E)$ is sufficiently close to the upper bound. We determine the nonemptiness, irreducibility and dimension of the Brill-Noether loci in the moduli spaces of sheaves with $h^0(E)$ in this range. When they are nonempty, these Brill-Noether loci are irreducible though almost always of larger than the expected dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brill-Noether theory on the projective plane for bundles with many sections Coskun, Izzet Huizenga, Jack Raha, Neelarnab Algebraic Geometry Primary: 14J60, 14J26. Secondary: 14D20 The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $E$ be a semistable sheaf on the projective plane. In this paper, we give an upper bound for $h^0(E)$ in terms of the rank $r$ and the slope $μ$ of $E$. We show that the bound is achieved precisely when $E$ is a twist of a Steiner bundle. We classify the sheaves $E$ such that $h^0(E)$ is sufficiently close to the upper bound. We determine the nonemptiness, irreducibility and dimension of the Brill-Noether loci in the moduli spaces of sheaves with $h^0(E)$ in this range. When they are nonempty, these Brill-Noether loci are irreducible though almost always of larger than the expected dimension. |
| title | Brill-Noether theory on the projective plane for bundles with many sections |
| topic | Algebraic Geometry Primary: 14J60, 14J26. Secondary: 14D20 |
| url | https://arxiv.org/abs/2409.18008 |