A Simple Remark on Fields of Definition
Keywords:Algebraic curves, field of moduli, field of definition, curvas algebraicas, campo de módulos, campo de definición.
Let K< L be an extension of fields, in characteristic zero, with L algebraically closed and let K< L be the algebraic closure of K in L. Let X and Y be irreducible projective algebraic varieties, X defined over K and Y defined over L, and let π : X → Y be a non-constant morphism, defined over L. If we assume that K ≠ L,then one may wonder if Y is definable over K. In the case that K = Q, L = C and that X and Y are smooth curves, a positive answer was obtained by Gonzalez-Diez. In this short note we provide simple conditions to have a positive answer to the above question. We also state a conjecture for a class of varieties of general type.
 G. González-Diez. Variations on Belyi’s Theorem. Quart. J. Math. 57 (2006), 339-354.
 H. Hammer and F Herrlich. A Remark on the Moduli Field of a Curve. Arch. Math. 81 (2003), 5-10.
 I. Tsai. Dominating the varieties of general type. J. Reine Angew. Math. 483 (1997), 197-219.