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Gröbner strata in the Hilbert scheme of points. (English) Zbl 1237.14012

Summary: We provide a framework for working with Gröbner 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 Gröbner 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 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.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Software:

Macaulay2
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Full Text: DOI arXiv Euclid

References:

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