2009/07/02 by Lederer, Mathias
#13F20 #13P10 #14C05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.0907.0302
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.