TITLE: Is rationality a deformation invariant property for smooth threefolds?
QUESTION [7 upvotes]: 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 $\mathbb{P}^3_{\mathbb{C}}$?

REPLY [6 votes]: 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) 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).