Free curves in Fano hypersurfaces must have high degree
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- by Raymond Cheng;
- Proc. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/proc/17206
- Published electronically: April 14, 2025
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Abstract:
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.References
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Bibliographic Information
- Raymond Cheng
- Affiliation: Institute of Algebraic Geometry, Leibniz University Hannover, Germany
- MR Author ID: 1298722
- ORCID: 0000-0002-6937-2422
- Email: cheng@math.uni-hannover.de
- Received by editor(s): June 14, 2024
- Received by editor(s) in revised form: July 26, 2024, July 31, 2024, December 2, 2024, December 6, 2024, and January 4, 2025
- Published electronically: April 14, 2025
- Additional Notes: The author was supported by a Humboldt Research Fellowship during the preparation of this note.
- Communicated by: Rachel Pries
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
- MSC (2020): Primary 14M22, 14J70; Secondary 14G17, 14J45
- DOI: https://doi.org/10.1090/proc/17206