Universal properties of the Corrado Segre embedding

Abstract

Let S(Π0,Π1) be the product of the projective spaces Π0 and Π1, i.e. the semilinear space whose point set is the product of the point sets of Π0 and Π1, and whose lines are all products of the kind {P0}×g1 or g0×{P1}, where P0, P1 are points and g0, g1 are lines. An embedding χ : S(Π0,Π1) → Π′ is an injective mapping which maps the lines of S(Π0,Π1) onto (whole) lines of Π′. The classical embedding is the Segre embedding, γ0 : S(Π0,Π1)→ Π. For each embedding χ, there exist an automorphism α of S(Π0,Π1) and a linear morphism ψ : Π→ Π′ (i.e. a composition of a projection with a collineation) such that χ = αγ0ψ. (Here αγ0ψ maps P onto ψ(γ0(α(P ))) =: Pαγ0ψ.) As a consequence, every S(Π0,Π1) which is embedded in a projective space is, up to projections, a Segre variety.