Uniform bounds for fields of definition in projective spaces

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 $k^{\prime }/k$ of degree $\le C_{n-1}$ such that $x$ lifts to a $k^{\prime }$-rational point of any resolution of singularities.