On the Maximum Gonality of a Curve over a Finite Field
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913896702410752 |
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| author | Faber, Xander Grantham, Jon Howe, Everett W. |
| author_facet | Faber, Xander Grantham, Jon Howe, Everett W. |
| contents | The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$. Over finite fields, a result of F.K. Schmidt from the 1930s can be used to prove that the gonality is at most $g+1$. Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g \ge 5$ over a finite field, the gonality is at most $g$. For genus $g = 3$ and $g = 4$, the same result holds with exactly $217$ exceptions: There are two curves of genus $4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The genus-$4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-$3$ examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_14307 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Maximum Gonality of a Curve over a Finite Field Faber, Xander Grantham, Jon Howe, Everett W. Algebraic Geometry Number Theory The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$-morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \ge 2$ is at most $2g - 2$. Over finite fields, a result of F.K. Schmidt from the 1930s can be used to prove that the gonality is at most $g+1$. Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g \ge 5$ over a finite field, the gonality is at most $g$. For genus $g = 3$ and $g = 4$, the same result holds with exactly $217$ exceptions: There are two curves of genus $4$ and gonality $5$, and $215$ curves of genus $3$ and gonality $4$. The genus-$4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus-$3$ examples. |
| title | On the Maximum Gonality of a Curve over a Finite Field |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2207.14307 |