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

We give a generalisation of the duality of a zero-dimensional complete intersection for the case of one-dimensional almost complete intersections, which results in a 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/Jf to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in ℙ2(ℝ) of even degree is given.