  For every $n\ge 6$, we give an example of a finite subset of $\mathbb{P}^{2}$
of degree $n$ which does not descend to any Brauer-Severi surface over the
field of moduli. Conversely, for every $n\le 5$ we prove that a finite subset
of degree $n$ always descends to a $0$-cycle on $\mathbb{P}^{2}$ over the field
of moduli.
