Gröbner cells of punctual Hilbert schemes in dimension two

We begin with a comprehensive discussion of the punctual Hilbert scheme regular two-dimensional local ring in terms Gröbner cells. These schemes are most degenerate fibers Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role study smooth surfaces. They generally singular, but their cells affine spaces; they admit explicit parametrization due to Conca and Valla. use this obtain decomposition compactified Jacobians plane curve singularities, which is non-trivial even for generalized (principal ideals only). One applications topological invariance certain variants corresponding motivic superpolynomials analytic deformations quasi-homogeneous singularities some similar families.