Проективтік геометрияда Proj құрылысы

In algebraic geometry, Proj is a construction analogous to the spectrum of a ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory. In this article, all rings will be assumed to be commutative and with identity.

Let be a commutative graded ring, whereis the direct sum decomposition associated with the gradation. The irrelevant ideal of is the ideal of elements of positive degreeWe say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set, For brevity we will sometimes write for .

The essential property of for the above construction was the ability to form localizations for each prime ideal of This property is also possessed by any graded module over , and therefore with the appropriate minor modifications the preceding section constructs for any such a sheaf, denoted , of modules on This sheaf is quasicoherent by construction. If is generated by finitely many elements of degree (e. g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on arise from graded modules by this construction. The corresponding graded module is not unique.

If we let the base ring be , thenhas a canonical projective morphism to the affine line whose fibers are elliptic curves except at the points where the curves degenerate into nodal curves. So there is a fibrationwhich is also a smooth morphism of schemes (which can be checked using the Jacobian criterion).

The projective hypersurface is an example of a Fermat quintic threefold which is also a Calabi–Yau manifold. In addition to projective hypersurfaces, any projective variety cut out by a system of homogeneous polynomialsin variables can be converted into a projective scheme using the proj construction for the graded algebragiving an embedding of projective varieties into projective schemes.

Weighted projective spaces can be constructed using a polynomial ring whose variables have non standard degrees. For example, the weighted projective space corresponds to taking of the ring where have weight while has weight 2.

The proj construction extends to bigraded and multigraded rings. Geometrically, this corresponds to taking products of projective schemes. For example, given the graded ringswith the degree of each generator Then, the tensor product of these algebras over gives the bigraded algebrawhere the have weight and the have weight Then the proj construction giveswhich is a product of projective schemes. There is an embedding of such schemes into projective space by taking the total graded algebrawhere a degree element is considered as a degree element. This means the th graded piece of is the moduleIn addition, the scheme now comes with bigraded sheaves which are the tensor product of the sheaves whereand are the canonical projections coming from the injections of these algebras from the tensor product diagram of commutative algebras.

A generalization of the Proj construction replaces the ring S with a sheaf of algebras and produces, as the result, a scheme which might be thought of as a fibration of Proj's of rings. This construction is often used, for example, to construct projective space bundles over a base scheme.

If S has the additional property that is a coherent sheaf and locally generates S over (that is, when we pass to the stalk of the sheaf S at a point x of X, which is a graded algebra whose degree zero elements form the ring then the degree one elements form a finitely generated module over and also generate the stalk as an algebra over it) then we may make a further construction. Over each open affine U, Proj S(U) bears an invertible sheaf O(1), and the assumption we have just made ensures that these sheaves may be glued just like the above; the resulting sheaf on is also denoted O(1) and serves much the same purpose for as the twisting sheaf on the Proj of a ring does.

Let be a quasi coherent sheaf on a scheme The sheaf of symmetric algebras is naturally a quasi coherent sheaf of graded modules, generated by elements of degree 1. The resulting scheme is denoted by If is of finite type, then its canonical morphism is a projective morphism. For any , the fiber of the above morphism over is the projective space associated to the dual of the vector space over
If is a quasi coherent sheaf of graded modules, generated by and such that is of finite type, then is a closed subscheme of and is then projective over In fact, every closed subscheme of a projective is of this form.

As a special case, when is locally free of rank , we get a projective bundle over of relative dimension Indeed, if we take an open cover of X by open affines such that when restricted to each of these, is free over A, then

and hence is a projective space bundle. Many families of varieties can be constructed as subschemes of these projective bundles, such as the Weierstrass family of elliptic curves. For more details, see the main article.

Global proj can be used to construct Lefschetz pencils. For example, let and take homogeneous polynomials of degree k. We can consider the ideal sheaf of and construct global proj of this quotient sheaf of algebras This can be described explicitly as the projective morphism .