A Hilbert scheme of points on a scheme X parametrizes configurations of n points on X. If X is a non-singular surface, the Hilbert scheme X^[n] is smooth and maps to the symmetric power Sⁿ X by Hilbert-Chow morphism. Hilbert schemes are connected to resolutions of singularities, hyperkähler geometry, quiver varieties, representation theory, and much more.
In this talk, we will prove Göttsche's formula for computing the Betti numbers of the Hilbert scheme X^[n] of n points on a surface. We show how it is a corollary of the general theory of intersection cohomology and the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber. The talk is based on Nakajima's book on Hilbert schemes.