Vol. 5, No. 4, 2011

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Global descent obstructions for varieties

Jean-Marc Couveignes and Emmanuel Hallouin

Vol. 5 (2011), No. 4, 431–463
Abstract

We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having $Q$ as field of moduli, that admit models over every completion of $Q$, but have no model over $Q$.

Keywords
algebraic curve, global field, descent, field of definition, field of moduli, stack
Mathematical Subject Classification 2000
Primary: 11R34
Secondary: 12G05, 11G35, 14D22
Milestones
Received: 11 September 2009
Revised: 22 November 2010
Accepted: 19 February 2011
Published: 21 December 2011
Authors
 Jean-Marc Couveignes Institut de Mathématiques de Bordeaux Université Bordeaux I et CNRS 161, cours de la Libération F-33405 Talence France http://www.math.u-bordeaux1.fr/~couveign/ Emmanuel Hallouin Département de mathématiques et informatique Université de Toulouse le Mirail 5, allées Antonio Machado F-31058 Toulouse France http://www.math.univ-toulouse.fr/~hallouin/