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Consider a projective flat morphism $$ f\colon X\to Y $$ between normal varieties. Let's say over the complex numbers. The geometric fibers of $f$ are all irreducible.

I would like a criterion to detect the normality (or the non-normality) of the image of $f$. Maybe in term of the differential of $f$.

Thanks

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    $\begingroup$ If you look at mathoverflow.net/questions/76815/… I think it is likely that your hopes will be dashed. In essence, unless $f$ is flat, it's unlikely that there can be any criteria $\endgroup$
    – meh
    Commented Dec 17, 2014 at 2:01
  • $\begingroup$ so, let's take $f$ flat. I will edit the question $\endgroup$
    – Giulio
    Commented Dec 18, 2014 at 7:15
  • $\begingroup$ In the post you suggested, the author says something like: "if the fibres are reduced and irreducible, then the image of f is normal". Why is this true?? $\endgroup$
    – Giulio
    Commented Dec 18, 2014 at 7:33
  • $\begingroup$ the point is that "fibers reduced and irreducible then the image is normal' is false. $\endgroup$
    – meh
    Commented Dec 18, 2014 at 14:52
  • $\begingroup$ here mathoverflow.net/questions/76815/… there is a comment "It is obvious that if the fibers are reduced and connected then the variety is normal. In your case the fibers are obviously not reduced. – Alexander Braverman S" $\endgroup$
    – Giulio
    Commented Aug 17, 2015 at 11:20

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