  The present paper shall provide a framework for working with Gr\"obner bases
over arbitrary rings $k$ with a prescribed finite standard set $\Delta$. We
show that the functor associating to a $k$-algebra $B$ the set of all reduced
Gr\"obner bases with standard set $\Delta$ is representable and that the
representing scheme is a locally closed stratum in the Hilbert scheme of
points. We cover the Hilbert scheme of points by open affine subschemes which
represent the functor associating to a $k$-algebra $B$ the set of all border
bases with standard set $\Delta$ and give reasonably small sets of equations
defining these schemes. We show that the schemes parametrizing Gr\"obner bases
are connected; give a connectedness criterion for the schemes parametrizing
border bases; and prove that the decomposition of the Hilbert scheme of points
into the locally closed strata parametrizing Gr\"obner bases is not a
stratification.
