Gr\"obner strata in the Hilbert scheme of points

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.