  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.
