The homology of projective hypersurfaces is classically known for smooth hypersurfaces. Due to
results of Dimca the homology of a singular hypersurface with isolated singularities is related to the
homology of the smooth case as follows: the difference is concentrated in one dimension and
related to the direct sum of the Milnor lattices of the singular points.
In the talk we will treat 1- dimensional singularities. By using a one parameter smoothing of an n- dimensional hypersurface we can compare with a smooth hypersurface. We call this the vanishing homology of the smoothing. We will show that this (relative) homology is concentrated in two dimensions only: n+1 and n+2.
Moreover we will give precise information and bounds for the Betti numbers of the vanishing homology in terms of properties of the singular set, the generic transversal singularities, the 'special' non-isolated singularities and (if they occur) the isolated singularities.
As an example: the n+2 Betti number is bounded by the sum of (generic) transversal Betti numbers on each irreducible component of the 1-dimensional singular set. In several cases this Betti number is zero.
We discuss several examples.
This is joint work with Mihai Tibar, and a preprint can be found on http://arxiv.org/abs/1411.2640