the Fitting ideals ${\mathfrak F}^i_\nu(\phi)$ which are homogeneous ideals in the ring $R$ of implicit variables. However, for applications in geometric modeling, it is also interesting to get such a description of the singular locus of $\phi$ in the source space ${\mathbb P}^2$ in place of the target space ${\mathbb P}^3$. From a computational point of view, this also allows to reduce the number of ambient variables by one. For that purpose, we just have to pull-back our Fitting ideals through $\phi$. For all integers $i\geq 0$ and $\nu\geq \nu_0$ we define the ideals of $S$ $$\phi^{-1}({\mathfrak F}_\nu^i):=\phi^\sharp({\mathfrak F}_\nu^i).S$$

It is clear that for all $i$ we have $\phi^{-1}({\mathfrak F}_\nu^i) \subset I$, that is to say that the base points of $\phi$ are always contained in the support of $\phi^{-1}({\mathfrak F}_\nu^i)$ for all $i\geq 0$ and all $\nu\geq \nu_0$. By Lemma \[fibreP2\], this support is exactly the base points of $\phi$ if $i=\binom{\nu+2}{2}-1$. In other words we have $$\phi^{-1}\left({\mathfrak F}_\nu^{\binom{\nu+2}{2}-1} \right)=I, \hspace{.2cm} \forall \, \nu \geq \nu_0.$$

For all $\nu\geq \nu_0$, we clearly have $\phi^{-1}({\mathfrak F}_\nu^0)=(0) \subset S$. Notice that this behavior is similar to substituting the parameterization $\phi$ into an implicit equation of ${\mathscr S}$ which returns $0$. However, 