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Let $X,Y$ be finite type projective schemes over $\mathbb{C}$, and $f:X\rightarrow Y$ be a surjective morphism (but not an isomorphism). Suppose it is known that $Y$ is reduced, and the fibers of $f$ are reduced as well.

Are there any known results about $f$, which ensures that $X$ is also reduced?

There are similar results for connectedness, but I can't find anything for reducedness.

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    $\begingroup$ With these assumptions, the answer is no: take $X=Y\amalg Z$ where $Z$ is an non-reduced closed subscheme of $Y$. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jul 19, 2020 at 15:09

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In Lemma 1.4. of this article, it is proven that $f$ flat, $X$ pure dimensional and $Y$ irreducible ensures that $X$ is reduced in your case.

Maybe some of these requirments can be relaxed, I haven't thought really thought about it.

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