Higher codimension nef and effective cycles on the Hilbert scheme of 3 points in projective 3-space

Nef and effective cones of divisors have been the subject of much study. In contrast, their higher codimension analogues are much harder to compute and few examples exist in the literature. In this paper 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 ${\mathbb{P}}^3$. Our computation generalizes results of Ryan & Stathis and requires a careful analysis of the PGL orbits in the Hilbert scheme, as well as a new basis of the Chow ring inspired by Mallavibarrena and Sols.