that $h$ divides $h_\pp$ and completes the proof.

Notice that if $p_i\not= 0$, $g_i$ is a linear combination of the $g_j$’s for $j\not= i$ as $\ell_\pp (g_0,\ldots ,g_3)=0$. For instance if $p_0\not= 0$, $I=(\ell_\pp (f))+h(g_1,\ldots ,g_3)$.

\[rem:NoBPfinite\] The above lemma shows that fibers of dimension 1 can only occur when $V(I)\not= \emptyset$ as $V(I)\supseteq V((\ell_\pp (f),h_\pp ))$. It also shows that $$d\deg (h_\pp )\leq \deg (I)$$ if there is a fiber of dimension 1 of unmixed part given by $h_\pp$. Furthermore, any element $F$ in $I^{sat}$ of degree $<d$ has to be a multiple of $h_\pp$ for any fiber of dimension 1. As $h_\pp$ and $h_{\pp'}$ cannot have a common factor for $\pp \not= \pp'$, the product of these forms divide $F$. This gives a simple method to compute 1 dimensional fibers if such an $F$ exists.

\[thm:1dimfibers\] Let $\pp \in \operatorname{Proj}(R)$ be a $k$-rational closed point. If $\dim \pi_2^{-1}(\pp) = 1$ then, setting $\delta:=\deg (\pi_2^{-1}(\pp)))$, $$a^0 ( {\mathcal S}_I\otimes \kappa(\pp))\leq      \nu_0 -\delta -1$$ and $$\operatorname{reg}( {\mathcal S}_I\otimes \kappa(\pp))\leq    \max\{ \nu_0 -\delta -1,\operatorname{reg}(I)-d\}\leq \nu_0 -1.$$ As a consequence, $a^* ( {\mathcal S}_I\otimes \kappa(\pp))\leq   \nu_0 -2$.

By Lemma \[1fiber\], $I=(f)+h(g_1,g_2,g_3)$ with $h$ the equation of the unmixed part of the fiber. 