Hypersurface Singularities and Milnor Equisingularity

Abstract

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f,0}$, of $f$ at the origin, with either integral or $\mathbb{Z}/p\mathbb{Z}$ coefficients. If the critical locus of $f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f,0}$ completely determines if $f$ defines a family of isolated singularities, over a smooth base, with constant Milnor number. This result has a nice interpretation in terms of the structure of the vanishing cycles as an object in the perverse category.

A substantial improvement on our earlier paper, Hypersurface Singularities and the Swing. 15 pages