A picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909606980091904 |
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| author | Coppens, Marc |
| author_facet | Coppens, Marc |
| contents | Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve of genus $g$. This picture starts from irreducible components of $W^r_d(C)$ restricted to an open subset of $Pic (C)$ satisfying Brill-Noether theory as in the case of a general curve of genus $g$. We obtain some degeneracy loci associated to a morphism of locally-free sheaves on them of the expected dimension. All the irreducible components of the schemes $W^r_d(C)$ are translates of their closures in $Pic (C)$. We complete the proof that the schemes $W^r_d(C)$ are generically smooth in case $C$ is a general $k$-gonal curve (claimed but not completely proved before). We obtain some results on the tangent spaces to the splitting degeneracy loci for an arbitrary $k$-gonal curve and we obtain some new smoothness results in case $C$ is a general $k$-gonal curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21141 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$ Coppens, Marc Algebraic Geometry 14H51 Based on results on Hurwitz-Brill-Noether theory obtained by H. Larson we give a picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve of genus $g$. This picture starts from irreducible components of $W^r_d(C)$ restricted to an open subset of $Pic (C)$ satisfying Brill-Noether theory as in the case of a general curve of genus $g$. We obtain some degeneracy loci associated to a morphism of locally-free sheaves on them of the expected dimension. All the irreducible components of the schemes $W^r_d(C)$ are translates of their closures in $Pic (C)$. We complete the proof that the schemes $W^r_d(C)$ are generically smooth in case $C$ is a general $k$-gonal curve (claimed but not completely proved before). We obtain some results on the tangent spaces to the splitting degeneracy loci for an arbitrary $k$-gonal curve and we obtain some new smoothness results in case $C$ is a general $k$-gonal curve. |
| title | A picture of the irreducible components of $W^r_d(C)$ for a general $k$-gonal curve $C$ |
| topic | Algebraic Geometry 14H51 |
| url | https://arxiv.org/abs/2504.21141 |