  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.
