Discriminant of the ordinary transversal singularity type. The global equivalence class

Abstract

Consider a space X with the singular locus of positive dimension, Z=Sing(X). Suppose both Z and X are locally complete intersections at each point. The transversal type of X along Z is generically constant but at some points of Z it degenerates. In the previous work we have introduced (locally) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is "ordinary". We have established the basic local properties of the discriminant. In the current paper we consider the global case. We compute the equivalence class of the discriminant in the Picard group, Pic(Z). If $X$ is a hypersurface, the discriminant is naturally stratified by the singularities of fibres in the projectivized normal cone $\mathbb{P}$N$_{X/Z}$. In this case (under some additional assumptions) we compute the classes of low codimension strata in the Chow group, A$^2$(Z). As immediate applications, we (re)derive the multi-degrees of the classical discriminant of projective complete intersections and bound the jumps of multiplicity of X along Z (when the singular locus is one-dimensional).