Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in the family of K3 surfaces). It means that there exists a proper morphism $\mathcal{X}\rightarrow S$ with each fibre a K3 surface, $\mathcal{X}_{\eta}=X$ ($\eta$ a general point on $S$) and $\rho(\mathcal{X}_t)=1$ for some closed $t\in S$.

> I want to show that $S$ can be a smooth algebraic curve or better be $\mathbb{A}^1$.