A Simple Remark on Fields of Definition

Ruben A. Hidalgo

Resumen


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.

 


Palabras clave


Algebraic curves; field of moduli; field of definition; curvas algebraicas; campo de módulos; campo de definición.

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Referencias


M. de Franchis. Un teorema sulle involutioni irrationali. Rend. Circ. Mat. Palermo 36 (1913), 368.

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




DOI: http://dx.doi.org/10.4067/S0716-09172012000100003

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