The field of moduli of sets of points in P²

For every n≥ 6 n ≥ 6 , we give an example of a finite subset of P² P 2 of degree n which does not descend to any Brauer–Severi surface over the field of moduli. Conversely, for every n≤ 5 n ≤ 5 , we prove that a finite subset of degree n always descends to a 0-cycle on P² P 2 over the field of moduli.