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 ∈ I {\displaystyle L_{i},i\in I} on it is said to be an ample family if the open subsets U f = { f ≠ 0 } , f ∈ Γ ( X , L i ⊗ n ) , i ∈ I , n ≥ 1 {\displaystyle 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.2 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.3
A divisorial scheme has the resolution property; i.e., a coherent sheaf is a quotient of a vector bundle.4 In particular, a scheme that does not have the resolution property is an example of a non-divisorial scheme.
See also
- Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Théorie des Intersections et Théorème de Riemann-Roch. 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.
References
Borelli 1963, Introduction - Borelli, Mario (1963). "Divisorial varieties". Pacific Journal of Mathematics. 13 (2): 375–388. doi:10.2140/pjm.1963.13.375. MR 0153683. https://projecteuclid.org/euclid.pjm/1103035733 ↩
SGA 6, Proposition 2.2.3 and Definition 2.2.4. - Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Théorie des Intersections et Théorème de Riemann-Roch. 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. https://doi.org/10.1007%2FBFb0066283 ↩
Zanchetta 2020 - 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. https://www.sciencedirect.com/science/article/abs/pii/S0021869320300697 ↩
Zanchetta 2020, Just before Remark 2.4. - 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. https://www.sciencedirect.com/science/article/abs/pii/S0021869320300697 ↩