Hypersurface Singularities and Milnor Equisingularity

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, $F_{f, \mathbf 0}$,of$f$attheorigin,witheitherintegralor$\mathbb Z/p\mathbb Z$coefficients\mathrm{.}Ifthecriticallocusof$f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f,\mathbf{0}}$ 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.