TY - JOUR AU - Hidalgo, Ruben A. PY - 2012/01/29 Y2 - 2024/03/29 TI - A Simple Remark on Fields of Definition JF - Proyecciones (Antofagasta, On line) JA - Proyecciones (Antofagasta, On line) VL - 31 IS - 1 SE - DO - 10.4067/S0716-09172012000100003 UR - https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/1155 SP - 25-28 AB - <p><em>Let </em><span style="letter-spacing: 4.00pt;"><em>K&lt;</em></span><em> L be an extension of fields, in characteristic zero, with L algebraically closed and let </em><span style="letter-spacing: 3.50pt;">K&lt;</span> <span style="letter-spacing: 3.50pt;">L</span> <em>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 </em>: X → Y <em>be a non-constant morphism, defined over L. If we assume that K </em>≠ <em>L,then one may wonder if Y is definable over K. In the case that K </em>= <strong>Q, </strong>L = <strong>C </strong><em>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.</em></p><p> </p> ER -