7
$\begingroup$

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.

$\endgroup$
1

1 Answer 1

10
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.