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

## TL;DR

This paper constructs a smooth plane curve over the algebraic closure of rationals demonstrating that its field of moduli is not a field of definition, and that fields of definition differ from plane model-fields of definition, highlighting a novel phenomenon.

## Contribution

It provides the first known example of a smooth plane curve where the field of moduli is not a field of definition, and fields of definition differ from plane model-fields of definition.

## Key findings

- Field of moduli not a field of definition for the constructed curve
- Fields of definition do not coincide with plane model-fields of definition
- First example of this phenomenon in the literature

## Abstract

Given a smooth plane curve $\overline{C}$ of genus $g\geq 3$ 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 $C'$ defined over $L$ where $C'\times_L\overline{k}\cong \overline{C}$, and such that $C'$ is $L$-isomorphic to a non-singular plane model $F(X,Y,Z)=0$ in $\mathbb{P}^2_{L}$. {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.