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 |
Be the first to leave a comment!