On explicit descent of marked curves and maps
Abstract
We revisit a statement of Birch that the field of moduli for a marked threepoint ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.
 Publication:

arXiv eprints
 Pub Date:
 April 2015
 arXiv:
 arXiv:1504.02814
 Bibcode:
 2015arXiv150402814S
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory
 EPrint:
 35 pages