Free curves in Fano hypersurfaces must have high degree
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912949299314688 |
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| author | Cheng, Raymond |
| author_facet | Cheng, Raymond |
| contents | The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free curves in Fano hypersurfaces must have high degree Cheng, Raymond Algebraic Geometry 14M22, 14J70 (primary), 14G17, 14J45 (secondary) The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces. |
| title | Free curves in Fano hypersurfaces must have high degree |
| topic | Algebraic Geometry 14M22, 14J70 (primary), 14G17, 14J45 (secondary) |
| url | https://arxiv.org/abs/2404.17341 |