Segments and Hilbert schemes of points

Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order in the Hilbert scheme of points in Pn. In this context, we look into the properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also on the connections between the shape of generators of Borel ideals and the related Hilbert polynomial, thus providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.