On the nef and effective cones of Hilbert schemes of points in projective space

Abstract

We compute the nef cones in codimensions 2 & 3 and the effective cones in dimensions 2 & 3 for the Hilbert scheme of three points in $\mathbb{P}^3$. This involves studying the orbits of the PGL action on the Hilbert scheme, as well as extending Mallavibarrena and Sols' bases for the Chow groups of Hilbert schemes of points on $\mathbb{P}^2$ to the case of the Hilbert scheme of 3 points in $\mathbb{P}^3$. This work builds on results of Ryan and Stathis, and utilizes results of Rossello-Llompart on intersection products in the Hilbert scheme of three points in $\mathbb{P}^3$.