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
Remarks and variations: The construction can be viewed as a relative Proj of the symmetric algebra Sym
Applications: Projectivized bundles are fundamental in the study of projective bundles, Chern classes, and intersection theory;