Divisorial scheme

Not to be confused with Divisorial sheaf.

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).

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 $U_{f}=\{f\neq 0\},f\in \Gamma (X,L_{i}^{\otimes n}),i\in I,n\geq 1$ 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

• 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). Lecture Notes in Mathematics (in French). Vol. 225. 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 (2): 375–388. doi:10.2140/pjm.1963.13.375. MR 0153683.
• Zanchetta, Ferdinando (15 June 2020). "Embedding divisorial schemes into smooth ones". Journal of Algebra. 552: 86–106. doi:10.1016/j.jalgebra.2020.02.006. ISSN 0021-8693.