A remark on the field of moduli of Riemann surfaces
  Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let ${\rm Aut}(S)$ be its group of conformal automorphisms. It is well known that if either: (i) ${\rm Aut}(S)$ is trivial or (ii) $S/{\rm Aut}(S)$ is an orbifold of genus zero with exactly three cone points, then $S$ is definable over its field of moduli ${\mathcal M}(S)$. In the complementary situation, explicit examples for which ${\mathcal M}(S)$ is not a field of definition are known. We provide upper bounds for the minimal degree extension of ${\mathcal M}(S)$ by a field of definition in terms of the quotient orbifold $S/{\rm Aut}(S)$. 
Rube\'en A. Hidalgo
math.CV math.AG