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

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

2003-09-19

arxiv.org/abs/math/0309333v1

Mathematics

Algebraic Geometry

40 pages

Scientific paper

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.

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