Discriminant of the ordinary transversal singularity type. The local aspects

Abstract

Consider a space X with the singular locus, Z=Sing(X), of positive dimension. Suppose both Z and X are locally complete intersections. The transversal type of X along Z is generically constant but at some points of Z it degenerates. We introduce (under certain conditions) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is `ordinary'. The scheme structure of this discriminant is imposed by various compatibility properties and is often non-reduced. We establish the basic properties of this discriminant: it is a Cartier divisor in Z, functorial under base change, flat under some deformations of (X,Z), and compatible with pullback under some morphisms, etc. Furthermore, we study the local geometry of this discriminant, e.g. we compute its multiplicity at a point, and we obtain the resolution of its structure sheaf (as module on Z) and study the locally defining equation.