Let $H\_1$ and $H\_2$ be two planes in $\mathbb{P}^3$.Let $P$ be a set of $9$ points such that no three lie on a line. Suppose $H\_1$ contains 4 of them and $H\_2$ contains remaining $5$ points. Is it true that $P$ imposes independent conditions on quadrics ? 



Independent conditions imposed by points in different planes
Let $C'' \subset H\_2$ be the unique conic on $H\_2$ containing the 5 points and let 
$$
\{C'\_t\}\_{t \in \mathbb{P}^1} \subset H\_1
$$
be the pencil of conics containing the 4 points. Let 
$$
L = H\_1 \cap H\_2.
$$
The points impose independent conditions on quadrics if and only if there is no $t$ such that 
$$
C'\_t \cap L = C'' \cap L.\tag{\*}
$$
Indeed, any quadric passing through these points (except for $H\_1 \cap H\_2$) intersects the planes along conics $C'\_t \cup C''$ for some $t$, and it intersects the line $L$ along a 2-point scheme (jr contains $L$), so $(\*)$ follows. 


The converse follows from the exact sequences
$$
0 \to \mathcal{O}\_{\mathbb{P}^3} \to \mathcal{O}\_{\mathbb{P}^3}(2) \to \mathcal{O}\_{H\_1 \cup H\_2}(2) \to 0
$$
and
$$
0 \to \mathcal{O}\_{H\_1 \cup H\_2}(2) \to \mathcal{O}\_{H\_1}(2) \oplus \mathcal{O}\_{H\_2}(2) \to \mathcal{O}\_{L}(2) \to 0.
$$


