Let's fix a plane $\mathbb{P}^{2} \subset \mathbb{P}^{3}$ and $\gamma \in \mathbb{P}^{2}$
Let, $t \in \mathbb K$ ( where, $\mathbb K$ is the underlying closed field of characteristic $0$) ,then how can we choose a flat family of general points $\delta_t \in \mathbb{P}^{3}$ and a family of planes $H_t$ such that the following holds :
$(1)$ $\delta_t \in H_t$ for any $t$,
$(2)$ $\delta_t \notin \mathbb{P}^{2}$ for any $t \neq 0$
$(3)$ $H_0 = \mathbb{P}^{2}$ and $\delta_0 = \gamma \in \mathbb{P}^{2}$ ?
I know that for any point in $ \mathbb{P}^{3}$, it's contained in atleast $3$ hyperplanes, but how to construct such a flat family of points?
Can anyone give any references? Any help from anyone is welcome.
The standard construction, which gives a $\mathbb{P}^1$ family, which in turn of course will give an $\mathbb{A}^1$ family as you want is the following. Take any line $L$ through $\gamma$ not contained in your fixed plane. Fix a line $M$ in your plane not passing through $\gamma$. For each point $P\in L$, you get a plane which is the join of $P$ and $M$ and this gives such a family as you desire.