2022/10/10 by Giulio Bresciani, Bresciani, Giulio, Angelo Vistoli +1 · 1 citation
Computer Science · Mathematics · Medicine · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Magnolia and Illicium research #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2210.04789
openalex publication_date 2022/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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\inGal(k/k) such that the twist 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.