$\bullet$ Dualising once more, we obtain a double duality map $ I/I^2 {\longrightarrow}N^*$, where $N^*:=Hom_{\Sigma}(N,P/I)$. The kernel is again $\int I/I^2$, the cokernel $N^*/I:=N^*/(I/\int I)$ is a further invariant. For $I=(xy,xz,zx)$ one finds $\dim(N^*/I)=3$.\
$\bullet$ In the special where $\Sigma \subset ({{\mathbb C}}^3,0)$ is a space curve, $I$ is Cohen-Macaulay of codimension $2$. Such curves are syzygetic ($T_2(\Sigma)=0$) and unobstructed ($T^2(\Sigma)=0$). Furthermore, one has in that case $$\int I/I^2=Ext^1({\cal O}_{\Sigma},{\omega_{\Sigma}}),\;\;\; N^*/I=Ext^2({\cal O}_{\Sigma},{\omega_{\Sigma}})$$

$\bullet$ In [@dJ] and [@dJdJ] an invariant $VD_{\infty}(f)$ called the virtual number of $D_{\infty}$-points was defined. In case that $f \in I^2$ Pellikaan [@pellikaan2] (and more generally in [@dJvS1]), the expression of $f$ as a quadratic form in the generators of $I$ can be used to define a [*transverse Hessian map*]{} $$H: N \times N {\longrightarrow}{\cal O}_{\Sigma}$$ which defines by transposition a map $$h:  N {\longrightarrow}N^*$$ which has a finite cokernel in case $f$ has transverse $A_1$-singularities.

([@pellikaan2], [@dJvS1]) Let the radical ideal define a curve $\Sigma$ and let $f \in \int I$ have transverse type $A_1$. Assume furthermore that $\Sigma$ is smoothable, with $T_2(\Sigma)=T^2(\Sigma)=0$. Then: $$\dim(I/J_f)=c_{e,I}(f)+\dim(T^1(\Sigma))+VD_{\infty}(f)-\dim(\int I/I^2)$$ $$VD_{\infty}(f)=\dim(N/h(N))-\dim(N^*/I)+\dim(\int I/I^2)$$

These invariants have furthermore an interpretation in terms of deformation theory. An [*admissible deformation*]{} of the pair $f, 