Projective classical morphisms

A morphism $f:X\to Y$ of classical varieties is projective here if there is an integer $N\ge 0$ and a factorization $X\hookrightarrow Y\times {\mathbf{P}}_k^N\to Y$ in which the first map is a closed immersion and the second is projection. A closed immersion in this classical setting is an isomorphism onto a reduced closed subvariety. Morphisms have the general locally ringed-space meaning, checked on affine charts.

Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.