I am asking for a *reference* for the following lemma (for which I know a proof).



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> **Lemma.** Let $f\colon X\to Y$ be a surjective morphism of irreducible *smooth* complex algebraic varieties (separated, reduced, irreducible schemes of finite type over $\Bbb C$) with *smooth* fibres over closed points of $Y$. The $f$ is smooth if and only if all these fibres have the same dimension $\dim X-\dim Y$.
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Smoothness of a morphism of smooth varieties with smooth fibres
Let me expand my comment into an answer, so that the question will not appear as unanswered anymore.


If $f$ is smooth, then $f$ is flat and its fibres are automatically smooth and equidimensional, see [Vakil, Theorem. 25.2.2]. This provides one implication of your statement. The converse implication is [Vakil, Exercise 25.2.F (a)].


**References.**


[Vakil] R. Vakil, *The Rising Sea: Foundations of Algebraic Geometry*, November 18, 2017 [version](http://math.stanford.edu/%7Evakil/216blog/).


