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 $d > 2r − 2$. Let $\mathbb{P}^{r-1} \cong H \subset \mathbb{P}^{r}$ be a general hyperplane, such that the hyperplane section $H \cap C$ consists of $d$ points in *general position*.

Let $Q \in \Gamma(H,\mathcal{O}_H(2)) $  be a quadric in $H$ containing a hyperplane section $H \cap C$, or in other words, it's image/ "pullback" under canon restriction map $p^*_{H\cap C}$ in following diagram is zero:



$$
\require{AMScd}
\begin{CD}
\Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(2))   @>{p^*_{C,2}}  >> \Gamma(C, \mathcal{O}_{C}(2)) \\
@Vp^*_HVV  @VVV  \\
\Gamma(H, \mathcal{O}_{H}(2)) @>{p^*_{H\cap C}}>> \Gamma(H \cap C) =\mathbb{C}^d 
\end{CD}
$$



**Question:** Under which conditions it is known that the quadric $Q$ admits a lift to a quadric $\overline{Q} \in  \Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(2))$ containing the curve $C$, ie that which is mapped to zero under $p^*_{C,2}$ in the diagram?



**Motivation:** This can be regarded as generalization of the question posed [here][1], where is was asked - refering the a claim on page 75 in [these lecture notes][2] - why *projective normality* of $C$ gives a sufficient condition for the stated claim in the question. 

(Recall,  "*projectively normal*", 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.  
That looks cumbersome from viewpoint of naive dimension count. In general, the dimension of the kernel of  $p^*_{C,2}$ which is  $\dim_k \Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(k)) -\dim_k\text{Image}(p^*_{C,k})$ is bounded from below by $\dim_k \Gamma(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(k)) -\dim_k\Gamma(C,\mathcal{O}_C(k)) $, and as the restriction of $p^*_{C,2}$ to kernel of $p^*_{H}$ is injective by nondegeneracy of $C$, it looks more naturally from dimensional reasons considering the diagram above that we could find a lift of $Q$ that maps zero, if the kernel of $p^*_{C,2}$ is big, or $\dim_k\text{Image}(p^*_{C,2})$ is small, but for $p^*_{C,2}$ surjective, the dimension of image gets maximal.

That looks rather counter intuitive.  
So I'm woundering if *projective normality" really is a sufficient criterion for the existence of such lift containing the curve $C$, and if not, what is the "right" criterion. Does it has some "geometric" intuition?

  [1]: https://mathoverflow.net/questions/466360/lifting-of-quadrics-containing-hyperplane-section-for-projectively-normal-curves
[2]: https://math.uchicago.edu/~amathew/287y.pdf