Higher Multiplicities and Almost Free Divisors and Complete Intersections

In this book, the author considers a general class of nonisolated hypersurface and complete intersection singularities called 'almost free divisors and complete intersections', which simultaneously extend both the free divisors introduced by K. Saito and the isolated hypersurface and complete intersection singularities. They also include discriminants of mappings, bifurcation sets, and certain types of arrangements of hyperplanes, such as Coxeter arrangements and generic arrangements. Topological properties of these singularities are studied via a 'singular Milnor fibration' which has the same homotopy properties as the Milnor fibration for isolated singularities.The associated 'singular Milnor number' can be computed as the length of a determinantal module using a Bezout-type theorem. This allows one to define and compute higher multiplicities along the lines of Teissier's $\mu ^*$-constants. These are applied to deduce topological properties of singularities in a number of situations including: complements of hyperplane arrangements, various nonisolated complete intersections, nonlinear arrangements of hypersurfaces, functions on discriminants, singularities defined by compositions of functions, and bifurcation sets. It treats nonisolated and isolated singularities together. It uses the singular Milnor fibration with its simpler homotopy structure as an effective tool. It explicitly computes the singular Milnor number and higher multiplicities using a Bezout-type theorem for modules.