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 {{{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 (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 [math]\displaystyle{ L_i, i \in I }[/math] on it is said to be an ample family if the open subsets [math]\displaystyle{ U_f = \{ f \ne 0 \}, f \in \Gamma(X, L_i^{\otimes n}), i \in I, n \ge 1 }[/math] 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
References
- ↑ Borelli 1963, Introduction
- ↑ SGA 6, Proposition 2.2.3 and Definition 2.2.4.
- ↑ Zanchetta 2020
- ↑ Zanchetta 2020, Just before Remark 2.4.
- Berthelot, Pierre, ed (1971) (in fr). 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. 225. Berlin; New York: Springer-Verlag. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8.
- Borelli, Mario (1963). "Divisorial varieties". Pacific Journal of Mathematics 13 (2): 375–388. doi:10.2140/pjm.1963.13.375. https://projecteuclid.org/euclid.pjm/1103035733.
- Zanchetta, Ferdinando (15 June 2020). "Embedding divisorial schemes into smooth ones" (in en). 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.
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