A note on the Brill-Noether loci of small codimension in moduli space of stable bundles
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| Format: | Preprint |
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2025
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| _version_ | 1866909975083745280 |
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| author | Biswas, Pritthijit Iyer, Jaya NN |
| author_facet | Biswas, Pritthijit Iyer, Jaya NN |
| contents | Let $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{C}$. Let $M_{X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether subvariety $W^{1}_{X}(2,L)$ of $M_{X}(2,L)$ which parametrises stable vector bundles having at least two linearly independent global sections. In this article, for generic $X$ and $L$, we show that $W^{1}_{X}(2,L)$ is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when $g\geq 5$. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15749 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the Brill-Noether loci of small codimension in moduli space of stable bundles Biswas, Pritthijit Iyer, Jaya NN Algebraic Geometry 14D20 Let $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{C}$. Let $M_{X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether subvariety $W^{1}_{X}(2,L)$ of $M_{X}(2,L)$ which parametrises stable vector bundles having at least two linearly independent global sections. In this article, for generic $X$ and $L$, we show that $W^{1}_{X}(2,L)$ is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when $g\geq 5$. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface. |
| title | A note on the Brill-Noether loci of small codimension in moduli space of stable bundles |
| topic | Algebraic Geometry 14D20 |
| url | https://arxiv.org/abs/2505.15749 |