Projective scheme
A closed subscheme (cf. Scheme) of a projective space . In homogeneous coordinates
on
, a projective scheme is given by a system of homogeneous algebraic equations
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Every projective scheme is complete (compact in the case ); conversely, a complete scheme is projective if there is an ample invertible sheaf (cf. Ample sheaf; Invertible sheaf) on it. There are also other criteria of projectivity.
A generalization of the concept of a projective scheme is a projective morphism. A morphism of schemes is called projective (and
is called a scheme projective over
) if
is a closed subscheme of the projective fibre bundle
, where
is a locally free
-module. A composite of projective morphisms is projective. The projectivity of a morphism is preserved also under a base change; in particular, the fibres of a projective morphism are projective schemes (but not conversely). If a scheme
is projective and
is a finite surjective morphism, then
is also projective.
Any projective scheme (over ) can be obtained using the construction of the projective spectrum (cf. Projective spectrum of a ring). Restricting to the case of an affine base,
, suppose that
is a graded
-algebra with the
-module
being of finite type and generating the algebra
, and suppose that
is the set of homogeneous prime ideals
not containing
. Equipped with the natural topology and a structure sheaf, the set
is a projective
-scheme; moreover, any projective
-scheme has such a form.
References
[1] | D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) |
Comments
Given a vector bundle over
(or, equivalently, a locally free
-module
), the associated projective bundle, or projective fibre bundle, has as fibre over
the projective space
of all one-dimensional subspaces of the vector space
.
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 |
Projective scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_scheme&oldid=23935