Let $\pi: \mathcal{X} \to \mathbb{P}^1$ be a pencil of projective $\mathbb{C}$-varieties such that a general fiber is smooth. Let $\mathbf{P}$ be one of the properties: rational, unirational, stably rational. My questions is: is there a property $\mathbf{P}$ such that if a general fiber of $\pi$ satisfies $\mathbf{P}$ then so does $\mathcal{X}$? Can we say anything if we add an extra assumption that a general fiber of $\pi$ is isomorphic to a fixed smooth projective variety $F$ (i.e., $\pi$ is an isotrivial family over an open subset of $\mathbb{P}^1$ with each fiber isomorphic to $F$), satisfying property $\mathbf{P}$?



Rationality in pencil of projective varieties
For $\mathbf{P}=$ rational, the answer is no, even under your stronger assumption: take the Fermat cubic threefold $X\_0^3+\ldots+X\_4^3=0$, and consider the pencil of hyperplanes $X\_1=\lambda X\_0$. After blowing up the base locus you get a morphism $\mathscr{X}\rightarrow \mathbb{P}^1$ whose smooth fibers are isomorphic to the Fermat cubic surface, hence are rational, but $\mathscr{X}$ is not.


