Let $f:X \to Y$ be a finite surjective morphism of quasi-projective schemes over $\mathbb{C}$, $X$ is reduced and $Y$ is integral. Suppose that there exists an integer $n$ such that for every *closed* point $y \in Y$, the fiber $f^{-1}(y)$ is reduced and consists of $n$ distinct closed points. Is it true that $f$ is flat? 



Covering of schemes and flatness
Apply Hartshorne, Exercise II.5.8 to $f\_\*\mathcal{O}\_X$.


