Projectivized

Projectivization, in the context of algebraic geometry and related fields, refers to the construction that assigns to a vector bundle E over a base scheme or space X a new space P(E) that encodes lines (one-dimensional subspaces) in the fibers of E, or, equivalently, one-dimensional quotients of E. The precise interpretation depends on convention: many sources define P(E) as parameterizing 1-dimensional quotients of E, while others define it as parameterizing 1-dimensional subspaces of E or of E^*.

Construction and basic properties: Let X be a scheme and E a locally free sheaf of rank r on X. The projectivization is defined as P(E) = Proj(Sym^• E^*), together with a projection π: P(E) → X. There is a tautological line bundle O(1) on P(E) and a tautological quotient map π^* E → O(1) in Grothendieck’s convention (equivalently, a tautological subbundle O(-1) ⊂ π^* E in the alternate convention). The fiber of π over a point x ∈ X is the projective space P(E_x) of lines in the fiber E_x, hence P(E) is a fiber bundle with fiber P^{r-1}. If E is trivial, P(E) ≅ X × P^{r-1}.

Remarks and variations: The construction can be viewed as a relative Proj of the symmetric algebra Sym E^*, and it is the relative analogue of forming projective space from a vector space. In topological settings, a similar construction yields a projective bundle with fiber CP^{r-1}. The two conventions (quotients vs. subspaces) lead to slightly different identifications of the tautological line bundle but describe the same geometric object up to isomorphism.

Applications: Projectivized bundles are fundamental in the study of projective bundles, Chern classes, and intersection theory; they appear in the definition of Grassmannians, degeneracy loci, and moduli problems involving quotients of vector bundles.