S}_I)_\nu)$ depend on the integer $\nu$ (it is generated in degree $\nu$) whereas $\operatorname{ann}_R(({\mathcal S}_I)_\nu)=\ker(\phi^\sharp)$ for all $\nu\geq \nu_0$ (see [@BuJo03 Proposition 5.1] and [@BC05 Theorem 4.1]), where $\phi^\sharp:R\to S$ is the map of rings induced by $\phi$ (it sends each $x_i$ to $f_i(s)$ for all $i=0,\ldots,3$) and $\ker(\phi^\sharp)$ is the usual ideal definition of the image of $\phi$.

In this paper, we will push further the study of ${\mathfrak F}^0(({\mathcal S}_I)_\nu)$ by looking at the other Fitting ideals ${\mathfrak F}^i(({\mathcal S}_I)_\nu)$, $i>0$, since they provide a natural stratification : $${\mathfrak F}^0(({\mathcal S}_I)_\nu) \subset {\mathfrak F}^1(({\mathcal S}_I)_\nu) \subset {\mathfrak F}^2(({\mathcal S}_I)_\nu) \subset \cdots \subset  {\mathfrak F}^{\binom{\nu+2}{2}}(({\mathcal S}_I)_\nu)=R.$$ As we will see, these Fitting ideals are closely related to the geometric properties of the parameterization $\phi$. For simplicity, the Fitting ideals ${\mathfrak F}^i(({\mathcal S}_I)_\nu)$ will be denoted ${\mathfrak F}^i_\nu(\phi)$. We recall that ${\mathfrak F}^i_\nu(\phi) \subset R$ is generated by all the minors of size $\binom{\nu+2}{2}-i$ of any matrix representation $M(\phi)_\nu$.

Consider the following parameterization of the sphere $$\begin{aligned}
    \phi : {\mathbb P}^2_\CC & {\dashrightarrow}& {\mathbb P}^3_\CC \\
     (s_0:s_1:s_2) & \mapsto & (s_0^2+s_1^2+s_2^2: 2s_0s_2:2s_0s_1:s_0^2-s_1^2-s_2^2).\end{aligned}$$ Its matrix representations $M(\phi)_\nu$ have the expected properties for all $\nu\geq 1$ (see the next section). The computation 