The geometric interpretation of Fröberg–Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space

Abstract

The study of infinitesimal deformations of a variety embedded in projective space requires, at ground level, that of deformation 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 schemes. An optimal Hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of R. Fröberg and A. Iarrobino. We discuss these conjectures from a geometric point of view. They give, for each such collection, a function (based on dimension, number of points, and order of each neighbourhood) which should serve as an upper bound to its Hilbert function (Weak Conjecture). The Strong Conjecture predicts when the upper bound is sharp, in the case of equal order throughout. In general we refer to the equality of the Hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the Strong Hypothesis. We interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points. We develop techniques and insight toward the conjectures' verification and refinement. The main result gives an upper bound on the Hilbert function of a collection of infinitesimal neighbourhoods in Pn based on Hilbert functions of certain such subschemes of Pn−1. Further, equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand. It follows that an infinitesimal neighbourhood scheme obeys the Weak Conjecture provided that the schemes identified in codimension one satisfy the Strong Hypothesis. This observation is then applied to show that the Weak Conjecture does hold valid in Pn for n⩽3. The main feature here is that the result is obtained although the Strong Hypothesis is not known to hold generally in P2 and, further, P2 presents special exceptional cases. Consequences of the main result in higher dimension are then examined. We note, then, that the full weight of the Strong Conjecture (and validity of the Strong Hypothesis) are not necessary toward using the main theorem in the next dimension. We end with the observation of how our viewpoint on the Strong Hypothesis pertains to extra algebraic information: namely, on the structure of the minimal free resolution of an ideal generated by linear forms.