Gröbner strata in the Hilbert scheme of points

The present paper shall provide a framework for working with Gröbner bases over arbitrary rings $k$ with a prescribed finite standard set $Δ$. We show that the functor associating to a $k$-algebra $B$ the set of all reduced Gröbner bases with standard set $Δ$ 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 $Δ$ and give reasonably small sets of equations defining these schemes. We show that the schemes parametrizing Gröbner 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 Gröbner bases is not a stratification.

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