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

Abstract

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