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Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic fiber of $f$ is generically non-reduced. Does there exist a noetherian scheme $Z$ and a dominant morphism $Z \to Y$, locally of finite type such that:

1) $Z$ is reduced and $\dim Z=\dim Y$

2) the geometric generic fiber of the composition $(Z \times_Y X)_{\mathrm{red}} \to Z \times_X Y \to Z$ is generically reduced.

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  • $\begingroup$ See papers of Tsuji and Fujino $\endgroup$
    – user21574
    Commented Apr 30, 2016 at 21:47
  • $\begingroup$ @HassanJolany Do you have a title? $\endgroup$
    – Ron
    Commented Apr 30, 2016 at 21:59
  • $\begingroup$ see papers of Tsuji related to Ricci iteration $\endgroup$
    – user21574
    Commented Apr 30, 2016 at 22:01
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    $\begingroup$ Sorry, these articles do not seem to have anything to do with non-reduced schemes $\endgroup$
    – Ron
    Commented Apr 30, 2016 at 22:20
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    $\begingroup$ Yes, for any irreducible noetherian scheme $Y$ and $f:X \rightarrow Y$ of finite type. For $K = k(Y)$ and the perfect closure $K'/K$, $(X_{K'})_{\rm{red}}$ is geometrically reduced over $K'$ [EGA IV$_2$, 4.6.1, 4.6.2]. The radical of $O_{X_{K'}}$ is coherent, so for some $K$-finite $F \subset K'$ the $F$-scheme $(X_{F})_{\rm{red}}$ is geometrically reduced (see [EGA IV$_2$, 4.6.8]). Pick quasi-finite dominant $Z \rightarrow Y$ with $Z$ integral having function field $F/K$; the quasi-finiteness and dominance are a proxy for the desired dimension equality. $\endgroup$
    – nfdc23
    Commented May 1, 2016 at 22:05

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