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Let $X, Y$ be quasi-projective 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?

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The flatness of $f$ is equivalent to $X$ being a Severi-Brauer scheme over $Y$ (under your assumptions). See Grothendieck, "Le groupe de Brauer", Cor. 8.3. – Damian Rössler Sep 5 '13 at 10:18
up vote 9 down vote accepted

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

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