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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.

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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.

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    $\begingroup$ In fact such an example is not possible for curves, where rationality correspond to genus 0, and the genus is constant over flat families. $\endgroup$
    – IMeasy
    Jan 25, 2014 at 23:02
  • $\begingroup$ See Theorem 2 arxiv.org/pdf/1608.00951.pdf $\endgroup$
    – user21574
    Aug 17, 2016 at 1:22

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