Let $\Gamma \subset {\mathbb P}^2_{{\mathbb P}^3}$ be the graph of $\phi$ and $R_{+}=(X_0,\ldots,X_3)$ the graded maximal ideal of $R$. We have the following diagram $$\label{eq:diagram}
 \xymatrix@1{\Gamma\ \ar[d]_{\pi_1}\ar[rd]^{\pi_2}\ar@{^(->}[r] & {\mathbb P}^2_{{\mathbb P}^3_k} \\  {\mathbb P}^2_k & {\mathbb P}^3_k}$$ where all the maps are canonical. For any $\pp\in \operatorname{Proj}(R)$, we will denote by $\kappa(\pp)$ its residue field $R_\pp/\pp R_\pp$. The fiber of $\pi_2$ at $\pp\in \operatorname{Proj}(R)$ is the subscheme $$\pi_2^{-1}(\pp):=\operatorname{Proj}({\mathcal S}_I\otimes_R \kappa(\pp)) \subset {\mathbb P}^2_{\kappa(\pp)}.$$ For simplicity, we set $N_\nu^\pp:=\dim_{\kappa (\pp )}({\mathcal S}_I\otimes_{R} \kappa(\pp))_{\nu}$ for the Hilbert function of the fiber $\pi_2^{-1}(\pp)$ in degree $\nu$. It turns out that this quantity governs the support of the Fitting ideals ${\mathfrak F}^i_\nu(\phi)$ we are interested in.

\[propVanishingFitt\] For any $\pp\in \operatorname{Proj}(R)$ $${\mathfrak F}^i_\nu(\phi)\otimes_R \kappa(\pp)= 
\begin{cases}
    0 & \textrm{ if } \ 0\leq  i<N_\nu^\pp \\
    \kappa(\pp) & \textrm{ if } \  N_\nu^\pp \leq i
\end{cases}$$

First, as a consequence of the stability under base change of the Fitting ideals, one has $${\mathfrak F}^i(({\mathcal S}_I)_\nu)\otimes_{R} \kappa(\pp)={\mathfrak F}^i(({\mathcal S}_I)_\nu\otimes_{R} \kappa(\pp)).$$ Now, for any $\nu\geq 0$ the $\kappa(\pp)$-vector space $$({\mathcal S}_I)_\nu\otimes_{R} \kappa(\pp)=({\mathcal S}_I\otimes_{R} \kappa(\pp))_\nu$$ is of dimension $N_\nu^\pp$. The result then follows from the fact that the $i$-th Fitting ideals of a $\kappa(\pp)$-vector space of dimension $N$ is 