Gröbner Strata in the Punctual Hilbert Scheme

Given a standard set δ of a finite size r, we show that the functor associating to a k-algebra B the set of all reduced Gröbner bases with standard set δ is representable. We show that the representing scheme Hilb′ δ k[x]/k is a locally closed stratum in the punctual Hilbert scheme Hilbrk[x]/k. Moreover, we cover the punctual Hilbert scheme by open affine subschemes attached to all standard sets of size r, we describe the transition morphism between these open charts, we explicitly give the equations defining the image of Hilb′ δ k[x]/k in each of the charts, and we give a description of its closure.