  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.
