A transition matrix for two bases of the integral cohomology of the Hilbert scheme of points in the projective plane
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This work is devoted to comparing two integral bases for the integral cohomology of the Hilbert scheme of points in the projective plane. Let X be a smooth complex projective surface. One of the more interesting moduli spaces parameterizing objects associated with X is the Hilbert scheme of points, denoted X[superscript [n]], which parameterizes all 0-dimensional closed subschemes of length n in X. W. Wang, Z. Qin and W.P. Li used Heisenberg algebra operators to construct an integral basis of the integral cohomology of X[superscript [n]] whenever X is a smooth projective surface with vanishing odd Betti numbers. On the other hand, a work by G. Ellingsrud and S.A. Strømme gives a cellular decomposition of the Hilbert scheme of points on the projective plane. From this work, we have a second integral basis for the integral cohomology of X[superscript [n]] when X = P². We compare the elements of these two bases and ultimately give the upper triangular transition matrix from one basis to the other.