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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    $\begingroup$ 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. $\endgroup$ Sep 5, 2013 at 10:18

1 Answer 1


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


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