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 P^n. We prove that, for fixed n, there exists a constant C₍ such that every dynamical system P^n^n is defined over an extension of degree C₍ of the field of moduli. More generally, the same bound works for any kind of "algebraic structure" defined over P^n, 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 C₍-₁ such that x lifts to a k'-rational point of any resolution of singularities.