# Morphisms whose reduction is projective

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.

• 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. – Piotr Achinger Jul 11 at 9:43
• @PiotrAchinger is this possible if we require the reduction to be smooth Fano variety or smooth rational variety? – user142965 Jul 11 at 9:50
• 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. – Jason Starr Jul 11 at 10:01