NS(V) is finite is Francesco Severi's theorem of the base; the rank is the Picard number of V, often denoted ρ(V). Geometrically NS(V) describes the algebraic equivalence classes of divisors on V; that is, using a stronger, non-linear equivalence relation in place of linear equivalence of divisors, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to numerical equivalence, an essentially topological classification by intersection numbers. Relative Picard scheme Let f: X →S be a morphism of schemes. The relative Picard functor (or relative Picard scheme if it is a scheme) is given by: for any S-scheme T, where is the base change of f and fT * is the pullback. We say an L in has degree r if for any geometric point s → T the pullback of L along s has degree r as an invertible sheaf over the fiber Xs (when the degree is defined for the Picard group of Xs.) See also Sheaf cohomology Chow variety Cartier divisor Holomorphic line bundle Ideal class group Arakelov class group Group-stack Picard category Notes References Geometry of divisors Scheme theory Abelian varieties