Given a perfect field $k$ with algebraic closure $k'$ and a variety $X$ over $k'$, the field of moduli of $X$ is the subfield of $k'$ of elements fixed by elements $s$ of the Galois group of $k'$ over $k$ such that the twist $X_s$ is isomorphic to $X$. Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures.

In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety $X$ with a smooth marked point $p$ that ensure that the pair $(X,p)$ is defined over its field of moduli.