TITLE: Rationality in Families
QUESTION [7 upvotes]: 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.

REPLY [10 votes]: 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.