$V_{-}$ where $f<0$.

If $f \in {{\mathbb R}}[x,y,z]$ defines a curve with only ordinary double points, then $$Signature(B_{\phi}(f))=\chi(V_+)-\chi(V_{-})$$

The structure of the paper is as follows. After reviewing the duality in the complete intersection case, we explain the emergence of the Gorenstein module $I/J$ for one-dimensional almost complete intersections and explain how the pairing behaves in a relative situation. Then we revies some material about singularities with one-dimensional singular locus that provide a rich source of examples and explain for special situations the meaning of the resulting signature in the real case,

Duality for zero-dimensional complete intersections
===================================================

Let $P=K[[x_1,x_2,\ldots,x_n]]$ or any other regular local $K$-algebra of dimension $n$. Let there be given a sequence of elements $$f_1,\ldots,f_n \in P$$ and let $J$ be ideal generated by them. We denote the [*Koszul-complex*]{} associated to the sequence ${\bf f}:=f_1,f_2,\ldots,f_n$ by $$K_{\bullet}({\bf f}):=0 \rightarrow K_n \rightarrow K_{n-1} \rightarrow
 \cdots \rightarrow K_2 \rightarrow K_1 \rightarrow K_0 \rightarrow 0$$ Its terms are $K_p:= \bigwedge^p P^n$, so that $K_0=P$. The differentials are induced by sending the $i$-th basis vector $e_i$ of $P^n$ to $f_i \in P$. We denote its homology groups by $$H_k({\bf f}):=H_k(K_{\bullet}({\bf f}))$$ The following result is well-known (see e.g. [@brunsherzog], thm. 1.6.17 