The field of moduli of sets of points in P^2

Abstract For every n 6 n ≥ 6, we give an example of a finite subset of P^2 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 P^2 P 2 over the field of moduli.