  Hilbert schemes of zero-dimensional ideals in a polynomial ring can be
covered with suitable affine open subschemes whose construction is achieved
using border bases. Moreover, border bases have proved to be an excellent tool
for describing zero-dimensional ideals when the coefficients are inexact. And
in this situation they show a clear advantage with respect to Groebner bases
which, nevertheless, can also be used in the study of Hilbert schemes, since
they provide tools for constructing suitable stratifications.
  In this paper we compare Groebner basis schemes with border basis schemes. It
is shown that Groebner basis schemes and their associated universal families
can be viewed as weighted projective schemes. A first consequence of our
approach is the proof that all the ideals which define a Groebner basis scheme
and are obtained using Buchberger's Algorithm, are equal. Another result is
that if the origin (i.e. the point corresponding to the unique monomial ideal)
in the Groebner basis scheme is smooth, then the scheme itself is isomorphic to
an affine space. This fact represents a remarkable difference between border
basis and Groebner basis schemes. Since it is natural to look for situations
where a Groebner basis scheme and the corresponding border basis scheme are
equal, we address the issue, provide an answer, and exhibit some consequences.
Open problems are discussed at the end of the paper.
