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

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

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 $\dim X-\dim Y$.

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

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 of the same dimension $\dim X-\dim Y$ over closed points of $Y$. Then the morphism $f$ is smooth.

Feel free to migrate this elementary question to Mathematics StackExchange.

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

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 $\dim X-\dim Y$.

Feel free to migrate this elementary question to Mathematics StackExchange.

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

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. Then the morphism $f$ is smooth.

Feel free to migrate this elementary question to Mathematics StackExchange.

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

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 of the same dimension $\dim X-\dim Y$ over closed points of $Y$. Then the morphism $f$ is smooth.

Feel free to migrate this elementary question to Mathematics StackExchange.