2024/05/06 by Bresciani, Giulio
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2405.03621
We give a positive answer to a question of J. Doyle and J. Silverman about fields of definition of dynamical systems on ℙn. We prove that, for fixed n, there exists a constant Cn such that every dynamical system ℙn→ℙn is defined over an extension of degree ≤ Cn of the field of moduli. More generally, the same bound works for any kind of "algebraic structure" defined over ℙn, 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 ≤ Cn-1 such that x lifts to a k'-rational point of any resolution of singularities.