Divisorial scheme explained

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 (noetherian) 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

L _{i,} i \in I on it is said to be an ample family if the open subsets U _{f} = \{ f \ne 0 \}, f \in \Gamma(X, L ^{⊗ n} _{i} ), i \in I, n \ge 1 form a base of the (Zariski) topology on X; in other words, there is an open affine cover of X consisting of open sets of such form. A scheme is then said to be divisorial if there exists such an ample family of invertible sheaves.

Properties 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.

See also

• Jouanolou's trick

References

• Book: Berthelot . Pierre . Pierre Berthelot (mathematician) . . . Séminaire de Géométrie Algébrique du Bois Marie – 1966–67 – Théorie des intersections et théorème de Riemann–Roch – (SGA 6) (Lecture notes in mathematics 225) . Lecture Notes in Mathematics . 1971 . 225 . . Berlin; New York . fr . xii+700 . true . 10.1007/BFb0066283 . 978-3-540-05647-8 . 0354655 . .
• Borelli . Mario . Pacific Journal of Mathematics . 153683 . 375–388 . Divisorial varieties . 13 . 1963. 2 . 10.2140/pjm.1963.13.375 . free .
• Zanchetta . Ferdinando . Embedding divisorial schemes into smooth ones . Journal of Algebra . 15 June 2020 . 552 . 86–106 . 10.1016/j.jalgebra.2020.02.006 . en . 0021-8693.