Divisorial scheme

In algebraic geometry, a divisorial scheme is a scheme admitting an "ample family" of line bundles, as opposed to an ample line bundle. In particular, a quasi-projective variety is a divisorial scheme and the notion is a generalization of "quasi-projective". It was introduced in (in the case of a variety) as well as in (in the case of a scheme). The term "divisorial" refers to the fact that "the topology of these varieties is determined by their positive divisors." The class of divisorial schemes is quite large: it includes affine schemes, separated regular schemes and subschemes of a divisorial scheme (such as projective varieties).
Definition.
Here is the definition in SGA 6, which is a more general version of the definition of Borelli. Given a quasi-compact quasi-separated scheme "X", a family of invertible sheaves formula_1 on it is said to be an ample family if, for each formula_2 and each integer formula_3, the open subsets formula_4 form a base of the (Zariski) topology on "X"; in other words, those open sets are an open affine cover of "X". A scheme is then said to be divisorial if there exists such an ample family of invertible sheaves.