Isosingular loci of algebraic varieties

Abstract

We define the notion of isosingular loci of algebraic varieties, following the analytic case first studied by Ephraim. These are subsets of points where the variety has a prescribed formal singularity type. We show that the isosingular loci of an algebraic variety are locally closed in the Zariski topology and the associated reduced subschemes are smooth. Moreover, assuming characteristic 0, we prove the existence of a decomposition of the formal neighborhoods at closed points into a product of the respective isosingular locus at that point and a smooth factor. One of the main obstructions in the positive characteristic case is the non-separability of the orbit map associated to the contact group, as first observed by Greuel and Pham for isolated singularities.