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

Abstract

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.