  The study of infinitesimal deformations of a variety embedded in projective
space requires that of deformations of a collection of points, as specified by
a zero-dimensional scheme. Further, basic problems in infinitesimal
interpolation correspond directly to the analysis of such a scheme We interpret
conjectures of Froberg and Iarrobino on the Hilbert function of a general
collection of infinitesimal neighbourhoods of a collection of points in
projective space. A main result gives a method for verifying the Weak
Conjecture prescribed. Further, we present results on establishing the Strong
Conjecture, but also exhibit families of counterexamples, showing the need for
refinement of these conjectures.
