Fields of moduli and the arithmetic of tame quotient singularities

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 field automorphisms γ∈Gal(k¯¯¯/k) such that the Galois conjugate Xγ is isomorphic to X . The field of moduli is contained in all subextensions k⊂k′⊂k¯¯¯ such that X descends to k′ . In this paper, we extend the formalism and define the field of moduli when k is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove 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 a consequence, we prove that a variety X of dimension d with a smooth marked point p such that Aut(X,p) is finite, étale and of degree prime to d! is defined over its field of moduli.