Rank-2 wobbly bundles from special divisors on spectral curves
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914168937906176 |
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| author | Dinh, Duong |
| author_facet | Dinh, Duong |
| contents | We study rank-2 wobbly bundles on a Riemann surface $C$ of genus $g\geq 2$, i.e. semi-stable bundles admitting nonzero nilpotent Higgs fields, in terms of direct images of line bundles on smooth spectral curves $\tilde{C} \oversetπ{\rightarrow} C$. We give a sufficient condition for a semi-stable bundle $E$ to be wobbly: $E$ is a twist of $π_\ast \left(\mathcal{O}_{\tilde{C}}(\tilde{D}) \right)$ where the norm of $\tilde{D}$ is a summand of the divisor of a quadratic differential on $C$. We sketch the proof of the necessary condition statement, namely all rank-2 wobbly bundles can be characterised as such, and discuss how certain singularities of the wobbly locus arise from the Brill-Noether loci of spectral curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04224 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rank-2 wobbly bundles from special divisors on spectral curves Dinh, Duong Algebraic Geometry High Energy Physics - Theory Differential Geometry We study rank-2 wobbly bundles on a Riemann surface $C$ of genus $g\geq 2$, i.e. semi-stable bundles admitting nonzero nilpotent Higgs fields, in terms of direct images of line bundles on smooth spectral curves $\tilde{C} \oversetπ{\rightarrow} C$. We give a sufficient condition for a semi-stable bundle $E$ to be wobbly: $E$ is a twist of $π_\ast \left(\mathcal{O}_{\tilde{C}}(\tilde{D}) \right)$ where the norm of $\tilde{D}$ is a summand of the divisor of a quadratic differential on $C$. We sketch the proof of the necessary condition statement, namely all rank-2 wobbly bundles can be characterised as such, and discuss how certain singularities of the wobbly locus arise from the Brill-Noether loci of spectral curves. |
| title | Rank-2 wobbly bundles from special divisors on spectral curves |
| topic | Algebraic Geometry High Energy Physics - Theory Differential Geometry |
| url | https://arxiv.org/abs/2406.04224 |