Gr\"obner strata in the Hilbert scheme of points
Abstract
The present paper shall provide a framework for working with Gr\"obner bases over arbitrary rings k with a prescribed finite standard set . We show that the functor associating to a k-algebra B the set of all reduced Gr\"obner bases with standard set 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 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.