Real versus complex plane curves

Abstract

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve $C$ over an algebraically closed field $K$ of characteristic $0$ with field of moduli $k_C\subset K$ is defined by a polynomial with coefficients in $k^{\prime }$, where $k^{\prime }/k_C$ is an extension with $[k^{\prime }:k_C]\le 3$ and $[k^{\prime }:k_C]\mid \mathrm{deg}C$.

Contents are now part of arxiv:2303.01454