  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.
