# quasi-projective and separated as topological properties

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