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 (SAG 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."[1] 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 ${\displaystyle L_{i},i\in I}$ on it is said to be an ample family if, for each ${\displaystyle i\in I}$ and each integer ${\displaystyle n\geq 0}$, the open subsets ${\displaystyle U_{f}=\{f\neq 0\},f\in \Gamma (X,L_{i}^{\otimes n})}$ form a base of the (Zariski) topology on X; in other words, those open sets are an open affine cover of X.[2] A scheme is then said to be divisorial if there exists such an ample family of invertible sheaves.

References

1. Borelli 1963, Introduction
2. SGA 6, Definition 2.2.4. harvnb error: no target: CITEREFSGA_6 (help)

• Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie – 1966–67 – Théorie des intersections et théorème de Riemann–Roch – (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.
• Borelli, Mario (1963). "Divisorial varieties". Pacific Journal of Mathematics. 13: 375–388. MR 0153683.CS1 maint: ref=harv (link)