Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}(1) )\vert _C$ has degree $d$.
Assume that $d > 2r − 2$ (not sure if this assumption neccessary at this stage). Furthermore, assume that $C$ extremal and (consequently) projectively normal.
"Extremal" is a technical condition explained in these lecture notes and for purpose of this question boils down to assure that $\dim_k \Gamma(C,\mathcal{O}_C(2))=3r$ (see somewhere in the linked notes), and that $C$ is neccessarily "projectively normal", where latter means that the canonical pullback maps
$$ p_{C,k}:\Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(k))\to\Gamma(C,\mathcal{O}_C(k)) $$
are surjective for all $k$.
Let $\mathbb{P}^{r-1} \cong H \subset \mathbb{P}^{r-1}$ be a general hyperplane, such that the hyperplane section $H \cap C$ consists of $d$ points in general position.
Now the question: In the linked notes on page 75 (see from the "Let $d > 2r − 2$..." part) is claimed that under these assumptions
This means that any quadric $Q \in \Gamma(H,\mathcal{O}_H(2)) $ in $H$ containing a hyperplane section $H \cap C$ lifts to a quadric $\overline{Q} \in \Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(2))$ containing the curve $C$, ie that this lift $\overline{Q}$ maps to zero under the canonical pullback map $p_{C,2}$.
Why does this hold?
From elementary considerations, the preimage $p_H^{-1}(Q) $ of a $Q \in \Gamma(H,\mathcal{O}_H(2)) $ under canonical restriction $p_H:\Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(2)) \to \Gamma(H, \mathcal{O}_{H}(2))$ has as affine space dimension $r+1$, the kernel $p_{C,2}^{-1}(0) $ of the canonical pullback map $p_{C,2}$ above for $k=2$ has dimension $\binom{2+r}{r}-3r$ (here we used surjectivity).
But the sum of their codimensions (inside $\Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(2))$) doesn't exceed $\binom{2+r}{r}$, the dimension of $\Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(2))$, so naive counting argument not helps to find an element in the intersection of $p_H^{-1}(Q) $ and the kernel $p_{C,2}^{-1}(0) $.
So I not see why and if the answer is "yes", then how to see that the claimed lift of $Q$ containing whole $C$ exits.
