Hypersurface Singularities and Milnor Equisingularity

TL;DR

This paper establishes new bounds on the homology of Milnor fibers for hypersurfaces with singularities, and characterizes when such hypersurfaces form families of isolated singularities based on Betti numbers.

Contribution

It provides novel bounds for homology groups of Milnor fibers and characterizes families of isolated singularities through minimal Betti numbers.

Findings

New bounds on homology ranks of Milnor fibers with integral coefficients.

Characterization of isolated singularity families via minimal Betti numbers.

Interpretation of results in terms of vanishing cycles and perverse sheaves.

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, $F_{f, \mathbf 0}$, 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 $F_{f, \mathbf 0}$ 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.