Let $X\to S$ be a family of smooth projective complex threefolds over a connected base $S$, could it happen that for some $a, b\in S(\mathbb{C})$, the fiber $X\_a$ is birational to $\mathbb{P}^3\_{\mathbb{C}}$, while the fiber $X\_b$ is not birational to $\mathbb{P}^3\_{\mathbb{C}}$?



Is rationality a deformation invariant property for smooth threefolds?
I believe this is still an open question in dimension $3$.


There is an example due to Hassett, Kresch and Tschinkel ([Stable rationality in smooth families of threefolds](https://arxiv.org/abs/1802.06107)) of a smooth projective family of threefolds over a connected base where some fibers are *stably* rational and others are not.


Also in dimension at least $4$, there are several examples of families over a connected base where $X\_a$ is rational and $X\_b$ is not even stably rational (you can find references in the linked paper).


