Projective Algebraic Geometry III: Products, Graphs, Projections

V × W should be quasi-projective i.e., there is an isomorphism ψ: V ×W → ℙ ^K with ψ(V ×W) a quasi-projective variety in ℙ ^K ;

the notion should be local i.e., if ξ ∈ V, η ∈ W then there are open affine neighborhoods U _ξ , U _η of ξ, ξ respectively such that ψ(U _ξ ×U _η ) is open in ψ(U ×V) and ψ(U _ξ ×U _η ) ≃ U _ξ ×U _η (as a product of quasi-affine varieties); and,

the notion is categorical i.e., the projections π _V : V × W → V, π _W : V × W →W are regular, and, given Z with regular maps ϕ₁: Z → V, (ϕ₂: Z → W then there is a unique morphism ϕ ₁ ×ϕ ₂: Z → V ×W with π _V o (ϕ ₁ ×ϕ ₂) = ϕ ₁,π _W o (ϕ ₁ ×ϕ ₂) = ϕ ₂.