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


1. The general fiber is smooth.
2. 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>.



Rationality in Families
That's quite easy: take a family of cubic surfaces where the special fiber is a cone (for instance $X$ given by $X^3+Y^3+Z^3+uT^3=0$ in $\mathbb{A}^1\times \mathbb{P}^3$). Of course a cone over an elliptic curve is not rational.


