Let $\pi: X \to S$ be a flat, smooth family of genus $0$ curves, for example. Now take a point $x \in S$, then $x \in \pi{-1}(p)$ from some $p \in S$, i.e $x \in \mathbb{P}^1$.
I am trying to relate such families to first order infinitesimal deformations.
In particular, does there exist an open subset $ x\in U \subset X$ such that $U$ is a first order infinitesimal deformation of $\mathbb{P}^1$?