singularities appearing in a morsification.

(sketch) Consider a morsification over $S$. The construction of $I/J$ in the relative case \[relativemodule\] sheafifies in an obvious way to produce a sheaf ${\cal M}_S={\cal I}_S/{\cal J}_S$. Result \[flat\] implies that $\pi_*({\cal M}_S)$ is a free ${\cal O}_S$-module of finite rank, equal to $\dim(I/J)$. The freeness implies that this number is also equal to sum of the local contributions for the function $F_t$. A local calculation shows that both an $A_1$- and $D_{\infty}$-point give a contribution of $1$, hence the fomula follows.

In a similar vein, T. de Jong has shown that for a disentaglement of $f$ one has $VD_{\infty}(f)=\# D_{\infty}-2\# T$ where $T$ denotes the triple point $f=xyz$.

A signature theorem for functions
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We now consider a real function $f \in {{\mathbb R}}\{x_1,x_2,\ldots,x_n\}$ with jacobian ideal $J_f$ with $\dim(V(J_f))=1$. We let $I$ be the saturation of $J_f$ and consider the jacobian module $I/J_f$. Using the sequence ${\bf f}=f_1,f_2,\ldots,f_n$ and the isomorphism $\phi: P {\longrightarrow}\Omega^n$ given by $1 \mapsto dx_1\wedge dx_2 \wedge
\ldots dx_n$. From theorem \[pairing\] we obtain a non-degenerate pairing $$B_f: I/J_f \times I/J_f {\longrightarrow}{{\mathbb R}}$$

The signature invariant of $f$ is $$\sigma(f):=Signature(B_f) \in {{\mathbb Z}}$$

This is indeed an invariant 