On explicit descent of marked curves and maps
Abstract
We revisit a statement of Birch that the field of moduli for a marked three-point 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 e-prints
- Pub Date:
- April 2015
- DOI:
- 10.48550/arXiv.1504.02814
- arXiv:
- arXiv:1504.02814
- Bibcode:
- 2015arXiv150402814S
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Mathematics - Number Theory
- E-Print:
- 35 pages