$\phi^{-1}({\mathfrak F}_\nu^1)$ is nonzero since we have assumed $\phi$ birational. Actually, as a consequence of the results of Section \[sec:main\], the sequence of ideals $$\phi^{-1}({\mathfrak F}_\nu^1) \subset \phi^{-1}({\mathfrak F}_\nu^2) \subset \cdots \subset  \phi^{-1}\left({\mathfrak F}_\nu^{\binom{\nu+2}{2}-1} \right)=I$$ yields a filtration of ${\mathbb P}^2$ which is in correspondence with the singularities of $\phi$. Here is an illustrative example where we focus on the singularities of the parameterization $\phi$ in a situation where ${\mathscr S}$ is smooth and very simple.

Take four general linear forms $l_0,l_1,l_2,l_3$ in $S_1$ and consider the parameterization $$\phi: {\mathbb P}^2_\CC \rightarrow {\mathbb P}^3_\CC: (s_0:s_1:s_2) \rightarrow (l_1l_2:l_0l_3:l_0l_1:l_0l_1).$$ The closed image of $\phi$ is the plane of equation ${\mathscr S}: X_2-X_3=0$. There are three base points that are all local complete intersections: $$P_{01}=\{l_0=0\}\cap \{l_1=0\}, \ P_{02}=\{l_0=0\}\cap \{l_2=0\}, \ P_{13}=\{l_1=0\}\cap \{l_3=0\}.$$ The fiber of each base point by $\pi_1$ (see the diagram ) gives a line on ${\mathscr S}$, the projection of the associated exceptional divisor, say $D_{01}, D_{02}$ and $D_{13}$. The equations of the graph $\Gamma$ are $l_0X_0-l_2X_2,l_1X_1-l_3X_2,(l_3-l_2)X_2, X_2-X_3$ so that we have $$D_{01}: \{X_2=0\}\cap {\mathscr S}, \ D_{02}: \{l_1(P_{02})X_1-l_3(P_{02})X_2=0\}\cap {\mathscr S},$$ $$D_{13}: \{l_0(P_{13})X_0-l_2(P_{13})X_2=0\}\cap {\mathscr S}.$$ Now, let $P$ be a point on ${\mathscr S}$. If $P \notin D_{01}\cup D_{02} \cup 