The Proj Construction
Assume that is an integral graded ring, finitely-generated over ; such a ring corresponds to an irreducible affine variety , where the subscript of stands for the affine or inhomogeneous. is called the (weighted homogeneous) affine cone over the projective variety , which is defined as the homogeneous spectrum:
where the are the weights of the -action on the coordinates of .
In the case when (or any other field for that matter), and can be also viewed as .
When is generated by over , then is a quotient of , is a subspace of , and is a subspace of .
In particular, the inclusion of induces a map from , whose fibres are subvarieties of . For this reason, the variety is often referred to as being projective over (an) affine.
Consider the polynomial ring with the standard grading, so that the ideal in the definition is trivial. Then . Then
as the affine cone of -dimensional projective space is just -dimensional affine space .
Projective Scheme(s) over an Affine Scheme
(Hartshorne Corollary 5.16)
Let be a ring.
- If is a closed subscheme of , then there is a homogeneous ideal such that is the closed subscheme determined by .
- A scheme over is projective if and only if it is isomorphic to for some graded ring , where , and is finitely generated by as an -algebra.
Truncated Rings and the Veronese Embedding
Let be a graded ring. The -th truncated ring is the subring defined by
i.e we only consider the homogeneous pieces of that are of degree divisible by .
Let which has . Then , and , etc.
For a graded ring as above,
 M. Reid, ‘Graded rings and varieties in weighted projective space’.
 N. Proudfoot, ‘Lectures on Toric Varieties’.