Descent varieties for algebraic covers Pierre Dèbes†, Jean-Claude Douai†

Abstract. This paper is about descent theory for algebraic covers. Typical questions concern fields of definition, models, moduli spaces, families, etc. of covers. Here we construct descent varieties. Associated to any given cover f, they have the property that whether they have rational points on a given field k is the obstruction to descending the field of definition of f to k. Our constructions have a global version above moduli spaces of covers (Hurwitz spaces): here descent varieties are parameter spaces for Hurwitz families with some versal property. Descent varieties provide a new diophantine viewpoint on descent theory by reducing the questions to that of finding points on varieties. There are concrete applications. We answer a question raised in [DeHa] about totally p-adic models of covers. We also show that the subset of a given moduli space of covers of P1 where the field of moduli is a field of definition is Zariski-dense