A Brill-Noether Theorem for (toric) surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916997607981056 |
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| author | Cela, Alessio Lian, Carl |
| author_facet | Cela, Alessio Lian, Carl |
| contents | The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth, projective surface. Namely, a non-constant map deforms in a family of expected dimension as long as its image has anti-canonical degree at least 4.
In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms certain toric contractions. We raise the question of whether a Brill-Noether theorem could hold for toric varieties of higher dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Brill-Noether Theorem for (toric) surfaces Cela, Alessio Lian, Carl Algebraic Geometry The classical Brill-Noether theorem states that a map from a general curve to a projective space deforms in a family of expected dimension as long as its image does not lie in any hyperplane. In this note, we observe, as a direct consequence of standard results on Severi varieties, an analogous statement for maps from a general curve to any smooth, projective surface. Namely, a non-constant map deforms in a family of expected dimension as long as its image has anti-canonical degree at least 4. In the case of toric surfaces, curves of anti-canonical degree at most 3 admit a particularly elegant description in terms certain toric contractions. We raise the question of whether a Brill-Noether theorem could hold for toric varieties of higher dimension. |
| title | A Brill-Noether Theorem for (toric) surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2503.18905 |