10 - Slicing by pencils of hypersurfaces

Summary

Our aim is to extend the method of slicing by pencils to certain classes of nongeneric pencils. Nongeneric pencils may occur naturally in certain situations, for instance a distinguished class of nongeneric pencils is the class of polynomial functions f : ℂⁿ → ℂ.*

The setting of this chapter is general: a complex analytic space X = Y \ V with arbitrary singularities, where Y is some compact complex space and V is a closed analytic subspace, and a meromorphic function on Y. Considering pencils of hypersurfaces instead of pencils of hyperplanes, although not more general in itself, has the advantage to enfold the local theory of hypersurface singularities initiated by Milnor [Mi2], see Figure 9.1.

We have introduced in §9.1 the notion of singularity of a meromorphic function and we have seen that such singularities may occur in the axis. This means that the axis A of the pencil might not be anymore in general position in Y, as it is supposed to be in the classical Lefschetz theory. Pencils which allow isolated singularities in the axis turn out to be a natural class of “admissible” pencils, extending the class of generic Lefschetz pencils. Indeed, isolated singularities of functions on singular spaces are manageable enough objects, as we have already seen.

In the literature, examples of nongeneric pencils occurred sporadically; more recently they got into light, e.g. [KPS, Oka3], precisely because we can use nongeneric pencils towards more efficient computations.