modules $I/J_f$ were also considered by Ruud Pellikaan [@pellikaan1] under the name of [*jacobian module*]{}. The case where $I$ is a radical ideal defining a reduced curve, corresponds to $f$ having [*transverse type $A_1$*]{}. We proof the following theorem

Assume that $f \in {{\mathbb R}}\{x_1,x_2,\ldots,x_n\}$ has one-dimensional critical locus and transverse type $A_1$. Assume furthermore that either $f$ has a morsification or that $n=3$. Then

$$2 Signature(B_{\phi})=-\widetilde{\chi}(F_{-})+(-1)^{n-1}\tilde{\chi}(F_{+})$$ wher as before $F_{\pm}$ are the positive and negative Milnor fibres of $f$ and $\widetilde{\chi}$ denotes the reduced Euler characteristic.

Note that contrary to case of isolated singularities, the two Euler characteristics appearing at the right hand side are in general no longer equal up to a sign. It appears that the above statement has a much broader range of validity, and holds for many other transverse types of singularities. However, as stated, it is not true in full generality, but we were not able to identify the precise limits of its validity.\
A nice application arises in the situation of a homogeneous polynomial $f \in {{\mathbb R}}[x,y,z]$ of even degree. It defines a curve $C =\{f=0\} \subset {{\mathbb P}}^2({{\mathbb R}})$, whose complement consists of a part $V_{+}$ where $f>0$ and a part 