On the cup product for Hilbert schemes of points in the plane

We revisit Ellingsrud and Str{\o} mme's cellular decomposition of the Hilbert scheme of points in the projective plane. We study the product of cohomology classes defined by the closures of cells, deriving necessary conditions for the non-vanishing of cohomology classes. Though our conditions are formulated in purely combinatorial terms, the machinery for deriving them includes techniques from Bia{\l} ynicki-Birula theory: We study closures of Bia{\l} ynicki-Birula cells in complete complex varieties equipped with ample line bundles. We prove a necessary condition for two such closures to meet, and apply this criterion in our setting.