The geometry of Hilbert schemes of two points on projective space

TL;DR

This paper analyzes the geometric and cohomological properties of the Hilbert scheme of two points on projective space, providing bases for cohomology, cone computations, and applications to secant varieties.

Contribution

It introduces three cohomology bases, computes effective and nef cones, and applies these to secant varieties, advancing understanding of Hilbert schemes of two points.

Findings

Three bases for cohomology groups established

Effective and nef cones of higher codimensional cycles computed

Applications to degrees of secant varieties of complete intersections

Abstract

In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme. Next, we compute the class in one of these bases of the Chern classes of tautological bundles coming from line bundles. Finally, we provide an application of these results to the degrees of secant varieties of complete intersections.