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arXiv · 1504.02814

On explicit descent of marked curves and maps

Jeroen Sijsling·John Voight
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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 D\`ebes 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.

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Jeroen Sijsling, John Voight. 2015-04-10. On explicit descent of marked curves and maps. https://doi.org/10.1007/s40993-016-0057-3

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