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

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