Fields of Moduli of Smooth del Pezzo Surfaces

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 $X\simeq X^{\sigma }$. A fundamental question is under what conditions a variety admits a model over its field of moduli. We give a complete answer for smooth del Pezzo surfaces in characteristic $0$: every smooth del Pezzo surface of degree at least $3$ has a model over its field of moduli, whereas in degrees $1$ and $2$ there exist smooth complex del Pezzo surfaces with field of moduli $\mathbb{R}$ which do not admit a real model.