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Enriched Relative Polar Curves and Discriminants

David B. Massey
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Abstract

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 Lê'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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David B. Massey. 2006-07-26. Enriched Relative Polar Curves and Discriminants. https://arxiv.org/abs/math/0607210

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