On the Fields of Moduli of Curves of Genus Six

The field of moduli of a variety $X$ over an algebraically closed field $K$ is defined as the fixed field of those automorphisms $\sigma$ of $K$ for which $X\cong X^{\sigma }$. A fundamental question is under what conditions a variety admits a model over its field of moduli. We investigate this problem for curves of genus $6$, by considering the stratification of the moduli space $M_6$.