$0$ for $i<N$ and $\kappa(\pp)$ for $i\geq N$.

This leads us to focus on the behavior of the Hilbert functions $N_\nu^\pp$. By , it suffices to control the local cohomology of the fiber ${\mathcal S}_I\otimes \kappa(\pp)$ with respect to the ideal ${\mathfrak m}$. Part of this control is already known,

\[lemFromDimToN\] Let $\pp\in \operatorname{Proj}(R)$ and $v_\pp:=\dim ({\mathcal S}_I\otimes_R\kappa(\pp))$. Then, $$a^{v_\pp}({\mathcal S}_I\otimes_R\kappa(\pp))=a^{v_\pp} ({\mathcal S}_I\otimes_R R_\pp)\leq a^{v_\pp}({\mathcal S}_I) \\$$ and $$a^{v_\pp -1}({\mathcal S}_I\otimes_R\kappa(\pp))\leq \max\{ a^{v_\pp -1}({\mathcal S}_I\otimes_R R_\pp), a^{v_\pp} ({\mathcal S}_I\otimes_R R_\pp)\} 
\leq \max\{ a^{v_\pp -1}({\mathcal S}_I), a^{v_\pp} ({\mathcal S}_I)\} .$$

We are now ready to describe the support of the ideals ${\mathfrak F}^i_\nu(\phi)$ with $i\geq 0$ and $\nu\geq \nu_0$. We begin with the points on ${\mathscr S}$ whose fiber is 0-dimensional.

Let $\pp \in \operatorname{Proj}(R)$. If $\dim \pi_2^{-1}(\pp) = 0$ then for all $\nu\geq \nu_0$ $$\label{eqFittDeg}
    \pp \in V({\mathfrak F}^i_\nu(\phi)) \Leftrightarrow \deg(\pi_2^{-1}(\pp))\geq i+1.$$

By Proposition \[prop:regSym\] and Lemma \[lemFromDimToN\], we have $H^i_{\mathfrak m}({\mathcal S}_I\otimes_R\kappa(\pp))_\nu=0$ for all $\nu\geq \nu_0$ and $i\geq 0$. Then, the conclusion follows from Proposition \[propVanishingFitt\] and the equality since the Hilbert polynomial of the fiber $\pi_2^{-1}(\pp)$ is a constant polynomial which is equal to the degree of this fiber.

We now turn to points on ${\mathscr S}$ whose 