Plane model-fields of definition, fields of definition, the field of moduli of smooth plane curves

Abstract

Given a smooth plane curve $\overline{C}$ of genus $g\ge 3$ over an algebraically closed field $\overline{k}$, a field $L\subseteq \overline{k}$ is said to be a \emph{plane model-field of definition for $\overline{C}$} if $L$ is a field of definition for $\overline{C}$, i.e. $\exists$ a smooth curve $C^{\prime }$ defined over $L$ where $C^{\prime }{\times }_L\overline{k}\cong \overline{C}$, and such that $C^{\prime }$ is $L$-isomorphic to a non-singular plane model $F(X,Y,Z)=0$ in ${\mathbb{P}}_L^2$. {In this short note, we construct a smooth plane curve $\overline{C}$ over $\overline{\mathbb{Q}}$, such that the field of moduli of $\overline{C}$ is not a field of definition for $\overline{C}$, and also fields of definition do not coincide with plane model-fields of definition for $\overline{C}$.} As far as we know, this is the first example in the literature with the above property, since this phenomenon does not occur for hyperelliptic curves, replacing plane model-fields of definition with the so-called hyperelliptic model-fields of definition.