projectivization

Projectivization is a construction in algebraic geometry and differential geometry that associates to a vector bundle E over a base X a projective bundle P(E) over X. The construction is commonly defined using the symmetric algebra: if E is a locally free sheaf of rank r on X, then P(E) is Proj(Sym E^*), with a natural projection π: P(E) → X and a tautological line bundle O_{P(E)}(1) equipped with a universal surjection π^* E → O_{P(E)}(1). The fiber over a point x ∈ X is the projective space P(E_x), which is isomorphic to P^{r-1} over the residue field. Equivalently, P(E) parameterizes one-dimensional quotients of the fibers of E; dually, P(E^*) parameterizes lines in the fibers.

Properties and basic facts include that P(E) is locally isomorphic to X × P^{r-1}, making it a projective bundle over X. If E is the trivial bundle X × k^r, then P(E) ≅ X × P^{r-1}. The construction generalizes to real and complex manifolds, giving a fiber bundle with fiber RP^{r-1} or CP^{r-1}. The Picard group of P(E) satisfies Pic(P(E)) ≅ Z ⊕ Pic(X) (for r ≥ 2), generated by the tautological line bundle O(1) and the pullbacks of line bundles from X. The relative ample line bundle O(1) yields embeddings into projective spaces after twisting, and the Chow ring of P(E) can be described via Chern classes of E.

Applications of projectivization include the study of families of subspaces in a vector bundle, construction of moduli and parameter spaces, and computations in intersection theory. It also provides a natural geometric framework for the splitting principle and for expressing geometric properties of E through the geometry of P(E).