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Canonical Hilbert-Burch matrices for ideals of $k[x,y]$

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 Added by Conca Aldo
 Publication date 2007
  fields
and research's language is English




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An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Grobner cells and their Betti strata recovering results of Ellingsrud and Str{o}mme, Gottsche and Iarrobino.



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153 - Roser Homs , Anna-Lena Winz 2020
Sets of zero-dimensional ideals in the polynomial ring $k[x,y]$ that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Grobner cells. Conca-Valla and Constantinescu parametrize such Grobner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of $(x,y)$-primary ideals in Grobner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring $k[[x,y]]$ with a given leading term ideal with respect to a local term ordering.
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