On border basis and Groebner basis schemes

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 Gr\"obner 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 Gr\"obner basis schemes with border basis schemes. It is shown that Gr\"obner basis schemes and their associated universal families can be viewed as weighted projective schemes. A first consequence of our approach is the proof that {\it all}\/ the ideals which define a Gr\"obner 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 Gr\"obner basis scheme is smooth, then the scheme itself is isomorphic to an affine space. This fact represents a remarkable difference between border basis and Gr\"obner basis schemes. Since it is natural to look for situations where a Gr\"obner 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.