Canonical Hilbert-Burch matrices for ideals of $k[x,y]$
Aldo Conca, Giuseppe Valla
TL;DR
This paper introduces a canonical form for Hilbert-Burch matrices of Artinian ideals in two variables and applies it to analyze the structure of affine Gr"obner cells and Betti strata.
Contribution
It establishes a canonical Hilbert-Burch matrix for Artinian ideals in $k[x,y]$, enabling new insights into the geometry of Gr"obner cells and Betti strata.
Findings
Determined the dimension of certain affine Gr"obner cells.
Reproduced known results on Betti strata dimensions.
Provided a canonical form for Hilbert-Burch matrices.
Abstract
An Artinian ideal of has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Gr\"obner cells and their Betti strata recovering results of Ellingsrud and Str{\o}mme, G\"ottsche and Iarrobino.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Graph theory and applications · Algebraic Geometry and Number Theory