Complete classification of irreducible components of the Brill-Noether locus of rank-$2$ vector bundles of degree $d$ and speciality $2$ on a general $ν$-gonal curve
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915811500752896 |
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| author | Choi, Youngook Flamini, Flaminio Kim, Seonja |
| author_facet | Choi, Youngook Flamini, Flaminio Kim, Seonja |
| contents | This paper replaces the previous longer version and focuses on the specialty $2$ case. More precisely, in this paper we address the Brill-Noether theory for rank-two, degree $d$ stable bundles of speciality $2$ on a general $ν$-gonal curve $C$ of genus $g$, $3 \leq ν< \lfloor \frac{g+3}{2}\rfloor$, leveraging universal extension spaces, modular maps and recent developments in rank-one Brill-Noether theory over Hurwitz spaces on $C$.
We completely classify the irreducible components of such Brill-Noether loci in the whole range of interest for $d$, namely $2g-2 \leq d \leq 4g-4$. Using specialization techniques, we further uncover a stratification into locally closed subsets within some of these components, and we also provide additional insight into the birational geometry and the local structure of every such a component. Our methods yield descriptions of the irreducible components of any such Brill-Noether locus and, as a by-product of our more general results, also derive interesting consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant, rather than fixed degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete classification of irreducible components of the Brill-Noether locus of rank-$2$ vector bundles of degree $d$ and speciality $2$ on a general $ν$-gonal curve Choi, Youngook Flamini, Flaminio Kim, Seonja Algebraic Geometry 14H60, 14H10, 14D20, 14E08, 14J26, 14E05 This paper replaces the previous longer version and focuses on the specialty $2$ case. More precisely, in this paper we address the Brill-Noether theory for rank-two, degree $d$ stable bundles of speciality $2$ on a general $ν$-gonal curve $C$ of genus $g$, $3 \leq ν< \lfloor \frac{g+3}{2}\rfloor$, leveraging universal extension spaces, modular maps and recent developments in rank-one Brill-Noether theory over Hurwitz spaces on $C$. We completely classify the irreducible components of such Brill-Noether loci in the whole range of interest for $d$, namely $2g-2 \leq d \leq 4g-4$. Using specialization techniques, we further uncover a stratification into locally closed subsets within some of these components, and we also provide additional insight into the birational geometry and the local structure of every such a component. Our methods yield descriptions of the irreducible components of any such Brill-Noether locus and, as a by-product of our more general results, also derive interesting consequences for Brill-Noether loci of stable, rank-two bundles with a fixed general determinant, rather than fixed degree. |
| title | Complete classification of irreducible components of the Brill-Noether locus of rank-$2$ vector bundles of degree $d$ and speciality $2$ on a general $ν$-gonal curve |
| topic | Algebraic Geometry 14H60, 14H10, 14D20, 14E08, 14J26, 14E05 |
| url | https://arxiv.org/abs/2503.04132 |