Uniform bounds for fields of definition in projective spaces

TL;DR

This paper establishes uniform bounds on the degree of field extensions needed to define dynamical systems and algebraic structures on projective spaces, answering a longstanding question in algebraic dynamics.

Contribution

It proves the existence of a universal constant for fixed dimension that bounds the degree of fields over which dynamical systems and algebraic structures are defined.

Findings

Existence of a constant C_n bounding degrees for dynamical systems on P^n.

Uniform bounds apply to algebraic structures like curves and hypersurfaces.

Results extend to rational points on varieties with quotient singularities.

Abstract

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.