## intersection of hyperplane with a set of finite points in $\mathbb{P}^3$ which fails to impose independent conditions for sections of $\mathcal{O}(d)$

Let $$X$$ be a finite set of points in $$\mathbb{P}^3$$ of cardinality $$\ge 3d +3$$ which fails to impose independent conditions on sections of $$\mathcal{O}(d)$$ and $$X$$ does not pass through any quadratic hypersurface. Can we give an effective bound on cardinality of a subset $$Z \subset X$$ such that there exist a hyper plane containing $$Z$$ ?

If $$|X| < 3d + 2$$, then it is known that either there exist a line containing at least $$d+2$$ points of $$X$$ or there is a plane containing at least $$2d+2$$ points of $$X$$.

## isolated singularity of hypersurfaces in $\mathbb{P}^3$ along points in general position

Suppose we are ginen $$m$$ points in $$\mathbb{P}^3$$ in general position. Can we give an effective bound on $$m$$ such that there is no degree $$d$$ irreducible hypersurface having isolated singularity along the given $$m$$ points ?

## restricting sheaves in $\mathbb{P}^3$

Is it true that one has an exact sequence of the following form ? $$0 \to \mathcal{O}_Z \to I_{Z, \mathbb{P}^3}(1)\otimes \mathcal{O}_H \to I_{Z, H}(1) \to 0$$, where $$Z$$ is a finite set of points in $$\mathbb{P}^3$$ contained in a hyperplane $$H$$ and $$I_{Z, \mathbb{P}^3}$$ and $$I_{Z, H}$$ denote the ideal sheaf of $$Z$$ in $$\mathbb{P}^3$$ and $$H$$ respectively