  We study correspondences of zero-dimensional subschemes of a smooth variety X
"aligned" in the same way in inside the appropriate product of Hilbert schemes.
We show that such correspondences have etale covering by other such
correspondences that are affine bundles over flag bundles on the tangent bundle
of X. Moreover, we show that for many such coverings, the closures of the
correspondences having these coverings vary so much that there is no
compactification of the covering dominating all of these closures.
