Uniform bounds for fields of definition in projective spaces
| Title: | Uniform bounds for fields of definition in projective spaces |
|---|---|
| Authors: | Bresciani, Giulio |
| Publication Year: | 2024 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Number Theory, Mathematics - Algebraic Geometry, Mathematics - Dynamical Systems |
| More Details: | 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. |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2405.03621 |
| Accession Number: | edsarx.2405.03621 |
| Database: | arXiv |
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| Items | – Name: Title Label: Title Group: Ti Data: Uniform bounds for fields of definition in projective spaces – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Bresciani%2C+Giulio%22">Bresciani, Giulio</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2024 – Name: Subset Label: Collection Group: HoldingsInfo Data: Mathematics – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics+-+Number+Theory%22">Mathematics - Number Theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+-+Algebraic+Geometry%22">Mathematics - Algebraic Geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+-+Dynamical+Systems%22">Mathematics - Dynamical Systems</searchLink> – Name: Abstract Label: Description Group: Ab Data: 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. – Name: TypeDocument Label: Document Type Group: TypDoc Data: Working Paper – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="http://arxiv.org/abs/2405.03621" linkWindow="_blank">http://arxiv.org/abs/2405.03621</link> – Name: AN Label: Accession Number Group: ID Data: edsarx.2405.03621 |
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| RecordInfo | BibRecord: BibEntity: Subjects: – SubjectFull: Mathematics - Number Theory Type: general – SubjectFull: Mathematics - Algebraic Geometry Type: general – SubjectFull: Mathematics - Dynamical Systems Type: general Titles: – TitleFull: Uniform bounds for fields of definition in projective spaces Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Bresciani, Giulio IsPartOfRelationships: – BibEntity: Dates: – D: 06 M: 05 Type: published Y: 2024 |
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