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An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We showthat there is a canonical choice. As an application, we determine the dimensionof certain affine Gr\"obner cells and their Betti strata recovering results ofEllingsrud and Str{\o}mme, G\"ottsche and Iarrobino.

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