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 (Borelli 1963) (in the case of a variety) as well as in (SGA 6, Exposé II, 2.2.) (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 (noetherian) schemes and subschemes of a divisorial scheme (such as projective varieties).
01Definition
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 
02Properties and counterexample
Since a subscheme of a divisorial scheme is divisorial, "divisorial" is a necessary condition for a scheme to be embedded into a smooth variety (or more generally a separated Noetherian regular scheme). To an extent, it is also a sufficient condition.
A divisorial scheme has the resolution property; i.e., a coherent sheaf is a quotient of a vector bundle. In particular, a scheme that does not have the resolution property is an example of a non-divisorial scheme.
Sources and credits
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