Let $X, Y$ be quasiprojective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed $n$. Is it then true that $f$ is flat?

If $Y$ is nonreduced, then $X = Y_{red} \times \mathbf{P}^{n}$ is a counterexample. 

