Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity

Abstract

We describe the jacobian ideal of the fibers St of an equiresolvable deformation of a quasi-ordinary hypersurface singularity (S, 0). This kind of deformation, inspired by the work of Teissier, has generic _ber isomorphic to (S, 0) and special fiber a toric singularity. We show a formula, in terms of the logarithmic jacobian ideal, for the pull-back of the jacobian ideal of St in its normalization. The logarithmic jacobian ideal is studied in the normal toric case by Lejeune and Reguera in relation with the study of motivic invariants and arc spaces. We deduce some equisingularity properties of the normalized Nash modification of St.