Enriched Relative Polar Curves and Discriminants

Details Enriched Relative Polar Curves and Discriminants Enriched Relative Polar Curves and Discriminants

2006-07-07

arxiv.org/abs/math/0607210v2

Mathematics

Algebraic Geometry

34 pages, the revision contains a new section

Scientific paper

Let $(f, g)$ be a pair of complex analytic functions on a singular analytic space $X$. We give ``the correct'' definition of the relative polar curve of $(f, g)$, and we give a very formal generalization of L\^e's attaching result, which relates the relative polar curve to the relative cohomology of the Milnor fiber modulo a hyperplane slice. We also give the technical arguments which allow one to work with a derived category version of the discriminant and Cerf diagram of a pair of functions. From this, we derive a number of generalizations of results which are classically proved using the discriminant. In particular, we give applications to families of isolated ``critical points''.

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