Picard Scheme

The relative Picard scheme Pic_{X/S} (when representable) is the scheme (or group scheme) representing the sheaf that assigns to an S-scheme T the group of isomorphism classes of line bundles on X_T modulo pullback from T; it organizes families of line bundles and their algebraic equivalence classes and often decomposes into components such as Pic^0 and the Néron–Severi lattice.