(Projective)
\(f : X \to Y\) is projective if it factors as a closed immersion followed by the projection, \[ X \injects \PP^N_Y \to Y, \qquad \PP^N_Y \da \fiberprod{\PP^N_\ZZ}{\Spec \ZZ}{Y} , \] for some \(N\). It is quasi-projective if it factors as an open immersion into a scheme projective over \(Y\).
For \(Y\) Noetherian, projective implies proper. The proof combines properness of closed immersions, stability of properness under composition, and universal closedness of \(\PP^N_\ZZ \to \Spec \ZZ\) via elimination.
For a graded ring \(S\) with \(S_0 = A\) and \(S\) generated in degree one by finitely many elements, \(\Proj S \to \Spec A\) is projective. The blowup of a Noetherian scheme \(X\) along a closed subscheme with ideal sheaf \(\mci\) is the relative \(\Proj\) of the Rees algebra \(\bigoplus_{d\ge0}\mci^d\), and the blowup morphism is projective [@Har10a, Proposition II.7.13]; for \(X\) a projective variety, the blowup is again a projective variety.
A projective variety over an algebraically closed field \(k\) is the same as an integral closed subscheme of some \(\PP^n_k\), equivalently a closed subscheme that is reduced and irreducible. An arbitrary closed subscheme of \(\PP^n_k\) is a projective \(k\)-scheme but need not be a variety: \(V(x_0^2) \subseteq \PP^1_k\) is a nonreduced point, and \(V(x_0 x_1) \subseteq \PP^1_k\) is two points.