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 the revlex term order in the Hilbert scheme of points in ${\mathbb{P}}^n$. In this context, we look inside properties of several types of “segment” ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.