Let $X$ be a zero dimensional subscheme in $\mathbb{P}^3$ and $H$ be a hyperplane. Let $X^\prime$ be the residual subscheme with respect to $H$. Then there is an exact sequence of the form,
$0 \to \mathcal{I}_{X^\prime}(4) \to \mathcal{I}_X(5) \to \mathcal{I}_{X \cap H, H}(5) \to 0.$
Then cohomology sequence gives a map from $H^0(\mathcal{I}_{X \cap H, H}(5)) \to H^1(\mathcal{I}_{X^\prime}(4))$.
My question is what can we say about its image when $X\cap H$ imposes independent conditions on quintics on $H$ ? Does it remain same if we remove one point say, $p$ from $X \cap H$ and consider the image of $H^0(\mathcal{I}_{X \cap H \setminus p , H}(5))$ in $ H^1(\mathcal{I}_{X^\prime}(4))$ ?
