Subspace configurations and low degree points on curves
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866929567103451136 |
|---|---|
| author | Kadets, Borys Vogt, Isabel |
| author_facet | Kadets, Borys Vogt, Isabel |
| contents | This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the work of Harris-Silverman and Abramovich-Harris) that such curves, after a base change to $\overline{k},$ admit a map of degree at most $d$ onto $\mathbb{P}^1$ or an elliptic curve. For $d \geqslant 4$ the analogous statement was shown to be false by Debarre and Fahlaoui. We prove that once the genus of $X$ is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves with many low degree points, that were first discovered by Debarre and Fahlaoui. As an application of our methods, we obtain a classification of such curves over $k$ for $d=2,3$, and a classification over $\overline{k}$ for $d=4,5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_01067 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Subspace configurations and low degree points on curves Kadets, Borys Vogt, Isabel Number Theory Algebraic Geometry This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the work of Harris-Silverman and Abramovich-Harris) that such curves, after a base change to $\overline{k},$ admit a map of degree at most $d$ onto $\mathbb{P}^1$ or an elliptic curve. For $d \geqslant 4$ the analogous statement was shown to be false by Debarre and Fahlaoui. We prove that once the genus of $X$ is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves with many low degree points, that were first discovered by Debarre and Fahlaoui. As an application of our methods, we obtain a classification of such curves over $k$ for $d=2,3$, and a classification over $\overline{k}$ for $d=4,5$. |
| title | Subspace configurations and low degree points on curves |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2208.01067 |