Divisorial scheme
{{confuse|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 {{harv|Borelli|1963}} (in the case of a variety) as well as in {{harv|SGA 6|loc=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."{{harvnb|Borelli|1963|loc=Introduction}} 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 on it is said to be an ample family if the open subsets 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.{{harvnb|SGA 6|loc=Proposition 2.2.3 and Definition 2.2.4.}} 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.{{harvnb|Zanchetta|2020}}
A divisorial scheme has the resolution property; i.e., a coherent sheaf is a quotient of a vector bundle.{{harvnb|Zanchetta|2020|loc=Just before Remark 2.4.}} In particular, a scheme that does not have the resolution property is an example of a non-divisorial scheme.
See also
References
{{reflist}}
- {{cite book
| editor-last = Berthelot
| editor-first = Pierre
| editor-link = Pierre Berthelot (mathematician)
| editor2=Alexandre Grothendieck
| editor3=Luc Illusie
| title = Théorie des Intersections et Théorème de Riemann-Roch
| series = Lecture Notes in Mathematics
| year = 1971
| volume = 225
| publisher = Springer-Verlag
| location = Berlin; New York
| language = fr
| pages = xii+700
| no-pp = true
|doi=10.1007/BFb0066283
|isbn= 978-3-540-05647-8
| mr = 0354655
| ref ={{harvid|SGA 6}}
}}
- {{cite journal
| last = Borelli | first = Mario
| journal = Pacific Journal of Mathematics
| mr = 153683
| pages = 375–388
| title = Divisorial varieties
| url = https://projecteuclid.org/euclid.pjm/1103035733
| volume = 13
| year = 1963| issue = 2
| doi = 10.2140/pjm.1963.13.375
| doi-access = free
}}
- {{cite journal |last1=Zanchetta |first1=Ferdinando |title=Embedding divisorial schemes into smooth ones |journal=Journal of Algebra |date=15 June 2020 |volume=552 |pages=86–106 |doi=10.1016/j.jalgebra.2020.02.006 |url=https://www.sciencedirect.com/science/article/abs/pii/S0021869320300697 |language=en |issn=0021-8693}}
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