Gorenstein-duality for one-dimensional almost complete intersections-with an application to non-isolated real singularities

TL;DR

This paper extends Gorenstein duality from zero-dimensional complete intersections to one-dimensional almost complete intersections, analyzing signatures in real cases and relating them to Milnor fibers and Euler characteristics.

Contribution

It introduces a Gorenstein module framework for one-dimensional almost complete intersections and explores signature invariance under deformations, with applications to real singularities.

Findings

Signature of the pairing is constant under flat deformations.

Relates signature of Jacobian module to Euler characteristic of Milnor fibers.

Application to real algebraic curves of even degree.

Abstract

We give a generalization of the duality of a zero-dimensional complete intersection to the case of one-dimensional almost complete intersections, which results in a {\em Gorenstein module} $M=I/J$. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity $f$ with a one-dimensional critical locus, we relate the signature on the jacobian module $I/J_f$ to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in $\P^2(\R)$ of even degree is given.