Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913906416418816 |
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| author | Farkas, Gavril Feyzbakhsh, Soheyla Rojas, Andrés |
| author_facet | Farkas, Gavril Feyzbakhsh, Soheyla Rojas, Andrés |
| contents | We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz-Brill-Noether theory. In particular, we provide new proofs of the main non-existence and existence results in Hurwitz-Brill-Noether theory. Finally, using degree-k Halphen surfaces, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz-Brill-Noether theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19890 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions Farkas, Gavril Feyzbakhsh, Soheyla Rojas, Andrés Algebraic Geometry 14J28, 14H51, 18G10 We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz-Brill-Noether theory. In particular, we provide new proofs of the main non-existence and existence results in Hurwitz-Brill-Noether theory. Finally, using degree-k Halphen surfaces, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz-Brill-Noether theory. |
| title | Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions |
| topic | Algebraic Geometry 14J28, 14H51, 18G10 |
| url | https://arxiv.org/abs/2505.19890 |