Real versus complex plane curves

TL;DR

This paper investigates conditions under which complex plane curves can be defined over the real numbers, establishing a criterion based on isomorphism to their conjugates, and extends the analysis to algebraically closed fields of characteristic zero.

Contribution

It provides a characterization of real definability for complex plane curves of odd degree and generalizes the result to algebraically closed fields of characteristic zero.

Findings

Curves of odd degree are definable over the reals iff they are isomorphic to their conjugates.

Counterexamples exist for even degree curves.

Field of moduli extension bounds depend on the degree of the curve.

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 $k'$, where $k'/k_{C}$ is an extension with $[k':k_{C}]\le 3$ and $[k':k_{C}]\mid \operatorname{deg} C$.