Transverse-freeness in finite geometries
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916386229452800 |
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| author | Bruggemann, Charlie Choi, Vera Freidin, Brian Whyte, Jaedon |
| author_facet | Bruggemann, Charlie Choi, Vera Freidin, Brian Whyte, Jaedon |
| contents | We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in $n$-dimensional projective space that are tangent to every $k$-dimensional subspace, for some value of $n$ and $k$. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces we construct curves tangent to each of the lowest degree curves defined over the base field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transverse-freeness in finite geometries Bruggemann, Charlie Choi, Vera Freidin, Brian Whyte, Jaedon Algebraic Geometry 14N25, 14N20, 51E21, 51E23 We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in $n$-dimensional projective space that are tangent to every $k$-dimensional subspace, for some value of $n$ and $k$. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces we construct curves tangent to each of the lowest degree curves defined over the base field. |
| title | Transverse-freeness in finite geometries |
| topic | Algebraic Geometry 14N25, 14N20, 51E21, 51E23 |
| url | https://arxiv.org/abs/2409.05248 |