Uniform bounds for fields of definition in projective spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910436337647616 |
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| author | Bresciani, Giulio |
| author_facet | Bresciani, Giulio |
| contents | We give a positive answer to a question of J. Doyle and J. Silverman about fields of definition of dynamical systems on $\mathbb{P}^{n}$. We prove that, for fixed $n$, there exists a constant $C_{n}$ such that every dynamical system $\mathbb{P}^{n}\to\mathbb{P}^{n}$ is defined over an extension of degree $\le C_{n}$ of the field of moduli. More generally, the same bound works for any kind of "algebraic structure" defined over $\mathbb{P}^{n}$, such as embedded curves, hypersurfaces, algebraic cycles. As a consequence we prove that, if $x\in X(k)$ is a rational point of an $n$-dimensional variety with quotient singularities, there exists a field extension $k'/k$ of degree $\le C_{n-1}$ such that $x$ lifts to a $k'$-rational point of any resolution of singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03621 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform bounds for fields of definition in projective spaces Bresciani, Giulio Number Theory Algebraic Geometry Dynamical Systems We give a positive answer to a question of J. Doyle and J. Silverman about fields of definition of dynamical systems on $\mathbb{P}^{n}$. We prove that, for fixed $n$, there exists a constant $C_{n}$ such that every dynamical system $\mathbb{P}^{n}\to\mathbb{P}^{n}$ is defined over an extension of degree $\le C_{n}$ of the field of moduli. More generally, the same bound works for any kind of "algebraic structure" defined over $\mathbb{P}^{n}$, such as embedded curves, hypersurfaces, algebraic cycles. As a consequence we prove that, if $x\in X(k)$ is a rational point of an $n$-dimensional variety with quotient singularities, there exists a field extension $k'/k$ of degree $\le C_{n-1}$ such that $x$ lifts to a $k'$-rational point of any resolution of singularities. |
| title | Uniform bounds for fields of definition in projective spaces |
| topic | Number Theory Algebraic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2405.03621 |