On explicit descent of marked curves and maps

TL;DR

This paper explores conditions under which marked curves and maps can be explicitly descended to their field of moduli, extending classical criteria to wildly ramified cases and providing counterexamples for singular curves.

Contribution

It provides a constructive approach to descent of marked curves, extending classical criteria to more complex ramification scenarios and including explicit counterexamples.

Findings

Classical descent criteria can be extended to wildly ramified cases.

Explicit counterexamples are provided for singular curves.

A constructive method for descent based on algebraic branches is developed.

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.