  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.
