Intersection numbers of twisted cycles and cocycles for degenerate arrangements

TL;DR

This paper derives formulas for intersection numbers of twisted cycles and cocycles in degenerate hyperplane arrangements and explores their application to hypergeometric function relations.

Contribution

It provides explicit formulas for intersection numbers in degenerate arrangements and links these to hypergeometric function contiguity relations.

Findings

Formulas for intersection numbers in degenerate arrangements

Application to hypergeometric function contiguity relations

Enhanced understanding of twisted homology and cohomology in degeneracies

Abstract

We study the intersection numbers defined on twisted homology or cohomology groups that are associated with hypergeometric integrals corresponding to degenerate hyperplane arrangements in the projective $k$-space. We present formulas to evaluate the intersection numbers in the case when exactly one $(k+1)$-tuple of the hyperplanes intersects at a point. As an application, we discuss the contiguity relations of hypergeometric functions in terms of the intersection numbers on twisted cohomology groups.