Vanishing homology of projective hypersurfaces with 1-dimensional singularities

TL;DR

This paper investigates the vanishing homology of singular projective hypersurfaces with 1-dimensional singularities, revealing its concentration in two levels and calculating the ranks of key homology groups based on monodromy effects.

Contribution

It introduces the concept of vanishing homology for such hypersurfaces and determines the ranks of the nontrivial groups, highlighting the role of monodromy in their structure.

Findings

Vanishing homology concentrates in two levels for 1-dimensional singular loci.

Ranks of nontrivial homology groups depend on monodromy at special points.

The monodromy of the local system influences the homology structure.

Abstract

We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $\Sigma$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $\Sigma$ and on the effect of the monodromy of the local system over its complement.