I am looking for examples of flat, projective families $\pi:X\to B$ of schemes over $\mathbb C$ such that the general fiber $X_b$ is rational (i.e. birational to $\mathbb P^n$), while the special fiber $X_0$ is irrational. It is not known whether such a family exists with $\pi$ smooth. Instead, I would like to see examples where

- The general fiber is smooth.
- The special fiber is singular, irreducible.

Hassett-Pirutka-Tschinkel have produced smooth examples of the opposite phenomenon in https://arxiv.org/pdf/1603.09262.pdf.