$a^* ( {\mathcal S}_I\otimes \kappa(\pp))\leq \nu_0 -1$ and $\operatorname{reg}( {\mathcal S}_I\otimes \kappa(\pp))\leq  \nu_0$.\
(iii) If $\dim (\pi_2^{-1}(\pp))=1$ then $a^* ( {\mathcal S}_I\otimes \kappa(\pp))\leq \nu_0 -2$ and $\operatorname{reg}( {\mathcal S}_I\otimes \kappa(\pp))\leq  \nu_0-1$.

Let us now remark that there is no fiber of dimension two (i.e. equal to $ {\mathbb P}^{2}$) over a point $\pp \in \operatorname{Proj}(R)$. More generally :

\[FibreP2\] Let $\phi: {\mathbb P}^{n-1}_k {\dashrightarrow}{\mathbb P}^{m}_k$ be a rational map and $\Gamma$ be the closure of the graph of $\phi$. The following are equivalent : (i) $\pi : \Gamma \rightarrow   {\mathbb P}^m_k$ has a fiber $F_x$ equal to ${\mathbb P}^{n-1}_x$, (ii) $\phi$ is the restriction to the complement of a hypersurface of the constant map sending any point to the zero dimensional and $k$-rational point $x$.

For (i)$\Rightarrow$(ii), notice that $ {\mathbb P}^{n-1}_k\times \{ x\}$ is a subscheme of the absolutely irreducible scheme $\Gamma$ that has same dimension $n-1$. Hence $F_x =\Gamma$ showing (ii).

\[fibreP2\] For all $\nu\geq \nu_0$, $$\left\{\pp\in \operatorname{Proj}(R) : HP_{\pi_2^{-1}(\pp)}(\nu) = HP_{{\mathbb P}^2}(\nu) =\binom{\nu+2}{2} \right\}=\emptyset$$

It follows from Proposition \[prop:regSym\] that the defining ideal of the fiber is generated in degree at most $\nu_0$. As this ideal is non trivial for any point in $\operatorname{Proj}(R)$ by Lemma \[FibreP2\], 