2
$\begingroup$

Let $f : X\rightarrow Y$ be a dominant morphism between smooth projective varieties over an algebraically closed field $k$.

If $k$ has characteristic zero, then $f$ is generically smooth in the sense that there exists a non-empty open subset of $Y$ over which $f$ is smooth.

In general, if I assume furthermore that $f$ is proper and $f_*\mathcal O_X=\mathcal O_Y$, so that $f$ cannot factor through a purely inseparable morphism $Y'\rightarrow Y$, does generic smoothness still hold ?

$\endgroup$
  • 2
    $\begingroup$ No, this is not true. A typical eample are quasi-elliptic fibrations in characteristic $2$ or $3$. They are morphisms $f:X\to Y$ from a surface to a curve, satisfying your hypotheses, but whose general fibers are cuspidal curves of arithmetic genus $1$. $\endgroup$ – Olivier Benoist Sep 22 '15 at 6:33

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.