  It is a classical fact going back to F. Klein that an elliptic curve $E$ over
$\bar{\mathbb{Q}}$ is defined by a homogeneous polynomial in $3$ variables with
coefficients in $\mathbb{Q}(j_{E})$, where $j_{E}$ is the $j$-invariant of $E$,
and $\mathbb{Q}(j_{E})$ is the field of moduli of $E$. The general definition
of field of moduli goes back to T. Matsusaka and G. Shimura. With few
exceptions, it coincides with the intersection of the fields where the curve is
defined.
  We prove that every smooth plane curve of degree prime with $6$ is defined by
a homogeneous polynomial with coefficients in the field of moduli. Furthermore,
we show that most plane curves in arbitrary degree, and more generally most
algebraic cycles in $\mathbb{P}^{2}$ with finite automorphism group, descend to
a Brauer-Severi surface over the field of moduli.
