ALGEBRAIC COVERS - FIELD OF MODULI VERSUS FIELD OF DEFINITION

The field of moduli of a finite cover f : X --> B a priori defined ove r the separable closure K-s of a field K, with B defined over K, need not be a field of definition. This paper provides a cohomological meas ure of the obstruction. The case of G-covers, i.e., Galois covers give n together with their automorphisms, was fairly well-known. But no suc h cohomological measure was available for mere covers. In that situati on, the problem is shown to be controlled not by one, as for G-covers, but by several characteristic classes in H-2(K-m, Z(G)), where K-m is the field of moduli and Z(G) is the center of the group of the cover. Furthermore our approach reveals a more hidden obstruction coming on top of the main one, called the first obstruction and which does not e xist for G-covers. In contrast with previous works, our approach is no t based on well's descent criterion but rather on some elementary tech niques ib Galois cohomology. Furthermore the base space B can be an al gebraic variety of any dimension and the ground field K a held of any characteristic. Our main result yields concrete criteria for the field of moduli to be a field of definition. Our main result also leads to some local-global type results. For example we prove this local-to-glo bal principle: a G-cover f : X --> B is defined over Q if and only if it is defined over Q(p) for all primes p.