Normal crossing properties of complex hypersurfaces via logarithmic residues

TL;DR

This paper introduces a dual logarithmic residue map for hypersurface singularities, extending existing theorems to algebraically characterize normal crossing hypersurfaces and free divisors, with implications for Gorenstein singular loci.

Contribution

It develops a dual logarithmic residue map and provides an algebraic characterization of normal crossing hypersurfaces and free divisors, answering a question of Saito.

Findings

Extended a theorem of Lê and Saito to algebraic characterization

Characterized normal crossing divisors algebraically

Described free divisors with Gorenstein singular locus

Abstract

We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of L\^e and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, describe all free divisors with Gorenstein singular locus.