Uniform bounds for fields of definition in projective spaces

We give a positive answer to a question of J. Doyle and J. Silverman about fields of definition of dynamical systems on Pⁿ. We prove that, for fixed n, there exists a constant Cₙ such that every dynamical system Pⁿⁿ is defined over an extension of degree ≤ Cₙ of the field of moduli. More generally, the same bound works for any kind of "algebraic structure" defined over Pⁿ, such as embedded curves, hypersurfaces, algebraic cycles. As a consequence we prove that, if x∈ X(k) is a rational point of an n-dimensional variety with quotient singularities, there exists a field extension $k'/k$ of degree ≤ Cₙ₋₁ such that x lifts to a $k'$-rational point of any resolution of singularities.