Canonical Hilbert-Burch matrices for power series

Abstract

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 Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner 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 Gröbner 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.