Suppose I am looking to classify isomorphism classes of connected, compact, projective curves of genus zero. We usually start this moduli problem by considering a scheme $S$ and families of genus zero curves over $S$ up to isomorphism. Here $\mathbb{P}_1^k$ is projective over the algebraically closed field $k$. $S$ is assumed to be an affine $k$-algebra. I will define such a family over $S$ to be a smooth, proper morphism $\pi: X \to S$ of schemes of relative dimension $1$.
Now, we know genus zero curves over $\operatorname{Spec} k$ are isomorphic to the projective line $\mathbb{P}_k^1$.
Here is my question: Is such a family equivalent to a family of deformations of $\mathbb{P}_k^1$? That is, for the family $\pi: X \to S$, does there exist a point $s \in S$, such that $X_s \cong \mathbb{P}_k^1$