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Let $X$, $Y$ be irreducible Noetherian schemes. Let $f:X\rightarrow Y$ be a proper morphism. Assume that for any $y\in Y$, the base change $X\times \mathrm{Spec}\,k(y)\rightarrow \mathrm{Spec}\, k(y)$ is a smooth projective morphism. Is $f$ projective?

EDIT (in response to abx): assume moreover that $Y$ is reduced and geometrically unibranch.

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    $\begingroup$ No. In Faisceaux amples sur les schémas en groupes et les espaces homogènes (Springer Lecture Notes 119), §XII, Raynaud gives many examples of abelian schemes which are not projective. $\endgroup$
    – abx
    Apr 5, 2019 at 12:45

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