The field of moduli of sets of points in \(\mathbb {P}^{2}\)

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