  We work out details of the extrinsic geometry for two Hilbert schemes of some
contemporary interest: the Hilbert scheme of two points on the projective plane
and the dense open set parametrizing non-planar clusters in the punctual
Hilbert scheme of clusters of length four on affine three-space with support at
the origin. We find explicit equations in natural projective, respectively
affine embeddings for these spaces. In particular, we answer a question of
Bernd Sturmfels who asked for a description of the latter space that is
amenable to further computations. While the explicit equations we find are
controlled in a precise way by the representation theory of SL_3, our arguments
also rely on computer algebra.
