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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 ?

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    $\begingroup$ 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

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