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

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 Gr"obner cells and their Betti strata recovering results of Ellingsrud and Str{\o}mme, G"ottsche and Iarrobino.