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IIUC Remark 5.3.5 in EGA II says that there exist proper non-projective morphisms $X\rightarrow Y$ where $Y$ is the spectrum of a finite-dimensional $\mathbb{C}$-algebra such that the induced morphism $X_{red}\rightarrow Y_{red}$ is projective. Here is .pdf file.

May somebody give an example?

FWIW here some examples of finite-dimensional commutative associative unital $\mathbb{C}$-algebras are listed.

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    $\begingroup$ Take a K3 surface and deform it at random over $\mathbf{C}[t]/(t^n)$. Deformation theory will tell you that for a very general choice no ample line bundle will deform. $\endgroup$ – Piotr Achinger Jul 11 at 9:43
  • $\begingroup$ @PiotrAchinger is this possible if we require the reduction to be smooth Fano variety or smooth rational variety? $\endgroup$ – user142965 Jul 11 at 9:50
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    $\begingroup$ Welcome new contributor. Please confer Exercise III.5.9, p. 232, of Hartshorne's "Algebraic geometry". It can happen that $X_{\text{red}}$ is even projective space. $\endgroup$ – Jason Starr Jul 11 at 10:01

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