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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2508.00215 |
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| _version_ | 1866913969434787840 |
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| author | Gómez-Gonzáles, Claudio Wolfson, Jesse |
| author_facet | Gómez-Gonzáles, Claudio Wolfson, Jesse |
| contents | Let $k$ be a field of characteristic not 2 or 3. We establish polynomial lower bounds on the ambient dimension $N$ for an intersection $X\subset\mathbb{P}^N$ of quadrics, cubics and quartics to have a dense collection of solvable points, i.e. points in $X(k^{\mathsf{Sol}})$ where $k^{\mathsf{Sol}}/k$ is a solvable closure. Our method connects the classical theory of polar hypersurfaces, as redeveloped by Sutherland, to Fano varieties $\mathcal{F}(j,X)$ of $j$-dimensional linear subspaces on $X$, and we use this to obtain improved control on the arithmetic of $\mathcal{F}(j,X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvable points on intersections of quadrics, cubics, and quartics Gómez-Gonzáles, Claudio Wolfson, Jesse Algebraic Geometry Number Theory 14G05, 11D72, 14M15 Let $k$ be a field of characteristic not 2 or 3. We establish polynomial lower bounds on the ambient dimension $N$ for an intersection $X\subset\mathbb{P}^N$ of quadrics, cubics and quartics to have a dense collection of solvable points, i.e. points in $X(k^{\mathsf{Sol}})$ where $k^{\mathsf{Sol}}/k$ is a solvable closure. Our method connects the classical theory of polar hypersurfaces, as redeveloped by Sutherland, to Fano varieties $\mathcal{F}(j,X)$ of $j$-dimensional linear subspaces on $X$, and we use this to obtain improved control on the arithmetic of $\mathcal{F}(j,X)$. |
| title | Solvable points on intersections of quadrics, cubics, and quartics |
| topic | Algebraic Geometry Number Theory 14G05, 11D72, 14M15 |
| url | https://arxiv.org/abs/2508.00215 |