In algebraic geometry, a divisorial scheme is a scheme admitting an ample family of line bundles, as opposed to an
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
. In particular, a
quasi-projective variety In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in s ...
is a divisorial scheme and the notion is a generalization of "quasi-projective". It was introduced in (in the case of a variety) as well as in (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
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective spaces, projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n' ...
).
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. 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 In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smoo ...
(or more generally a separated Noetherian regular scheme). To an extent, it is also a sufficient condition.
A divisorial scheme has the
resolution property
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
; 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
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Jouanolou's trick
In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety ''X'', the existence of a surjection with affine fibers from an affine variety ''W'' to ''X''. The variety ''W'' is therefore homotopy-equivalent
In ...
References
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{{algebraic-geometry-stub
Algebraic geometry