Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ?

(By the way, projective implies a priori separated, but this is not true for quasi-projective, right?)

any reference is welcome thanks


No. If the reduction of a morphism is quasiprojective (or even projective), then the morphism itself need not be quasiprojective. See EGA II.5.3.5.

(Concerning your other question: Quasiprojective morphisms are separated. See EGA II.5.3.1.)


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