Proyecciones Journal of Mathematics Vol. 31, N^o 1, pp. 25-28, March 2012. Universidad Catolica del Norte Antofagasta - Chile

A simple remark on fields of definition *

Rubén Hidalgo

Universidad Técnica Federico Santa María, Chile

 

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ABSTRACT

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.

Keywords : Algebraic curves, field of moduli, field of definition.

Subjclass : [2000]14H30, 11G30, 14H25.

 

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REFERENCES

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

[2] G. Gonzalez-Diez. Variations on Belyi's Theorem. Quart. J. Math. 57 (2006), 339-354.         [ Links ]

[3] H. Hammer and F Herrlich. A Remark on the Moduli Field of a Curve. Arch. Math. 81 (2003), 5-10.         [ Links ]

[4] I. Tsai. Dominating the varieties of general type. J. Reine Angew. Math. 483 (1997), 197-219.         [ Links ]

Ruben A. Hidalgo

Departamento de Matemáatica, Universidad Tecnica Federico Santa María, Valparaíso,

Chile

e-mail : ruben.hidalgo@usm.cl

Received : April 2011. Accepted : November 2011

*Partially supported by projects Fondecyt 1110001 and UTFSM 12.11.01