The geometric interpretation of Froberg-Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space

Abstract

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.