Irreducible Subsets of Projective Space
Throughout this article, k is an algebraically closed field.
We wish to consider irreducible closed subsets of . For that we need the following preliminary result.
Let be a graded ring; a proper homogeneous ideal is prime if and only if:
Suppose is not prime so there exists such that . Since , among all homogeneous components of a pick of maximum degree such that ; similarly pick fo b so .
The degree-(d+e) component of ab is congruent to mod . Since is homogeneous we have , and by the given condition this means or , a contradiction. ♦
Prove that a proper homogeneous ideal of a graded ring A is primary if and only if
Suppose the closed subset corresponds to the homogeneous radical ideal , , where .
Then V is irreducible if and only if is prime.
(⇒) Suppose V is irreducible; let . If are homogeneous, then and are closed subsets of properly contained in V. Since V is irreducible the following shows :
(⇐) Let where is prime. Let be closed subsets with union V. Now write and for homogeneous radical ideals and . Then . Since is a homogeneous radical ideal, . By exercise B here, or . ♦
Let be a non-empty closed subset. Then is irreducible if and only if is irreducible.
By proposition 1, V is irreducible if and only if is prime. But (from an exercise here) so the result follows. ♦
Recall that a projective variety is a closed subset of some .
A quasi-projective variety is an open subset of a projective variety.
This merely defines it as a set; we need a geometric structure on it.
First, let be homogeneous polynomials of the same degree. If is such that not all , then we can define a function on an open subset U of containing as follows:
The map is well-defined: indeed if we can find an open neighbourhood U of such that . Also, if we replace projective coordinates with , then each where so
We write for the resulting function.
Let and be quasi-projective varieties and be a function.
We say is regular at if there is an open neighbourhood U of in V such that
for some homogeneous of the same degree.
We say is regular if it is regular at every , in which case we also say is a morphism of quasi-projective varieties.
From the above definitions, we obtain the category of all quasi-projective varieties and morphisms between them.
First consider the case where and are closed subsets.
E.g., let . A regular map in the earlier sense can be expressed as a polynomial , e.g. take . Via embeddings and taking and respectively, f can be written in terms of homogeneous coordinates as
since it is the homogenization of the map . This generalizes to an arbitrary regular map of closed subsets .
Conversely we have:
Let be regular under the new definition. Then there exist polynomials which represent .
We will prove this for the case where V is irreducible.
For each of , let be projection onto the i-th coordinate. Then is regular under the new definition, and by proposition 2 here (and its preceding discussion) can be represented as a polynomial . Hence we see that
for polynomials . ♦
Take the map given by
Note that the same set of polynomials works globally over the whole of .
Suppose . Let be the closed subset defined by . We define a map as follows
- Outside the point (1 : 1 : 0), take .
- Outside the point (1 : -1 : 0), take .
The map agrees outside those two points since due to the equality .
Consider the category of all quasi-projective k-varieties, with morphisms defined as above. Two such varieties are said to be isomorphic if they are isomorphic in the category.
A quasi-projective variety is said to be
- projective if it is isomorphic to a closed subset of some (this generalizes the existing definition of projective varieties);
- affine if it is isomorphic to an affine k-variety (closed subspace of some );
- quasi-affine if it is isomorphic to an open subset of an affine variety.
In example 3 above, we get an isomorphism since we have the reverse map
As an exercise, prove that and .
The coordinate ring of a quasi-projective variety V is the set
taken to be a k-algebra via point-wise addition and multiplication:
As before, a regular map of quasi-projective varieties induces a ring homomorphism . By lemma 2, when V is affine agrees with our earlier version (we proved this in the case where V is irreducible).
For each , we have an automorphism
Note that for . Also if and only if g is a scalar multiple of the identity matrix, so we get an injective homomorphism . In fact this is an isomorphism of groups.
E.g. when n = 1, we get the Möbius transformations:
We have an isomorphism between the quasi-affine variety and via the maps
Hence is an affine variety even though it is not closed in . From the isomorphism we also have:
Let . We will show that V is not affine. Indeed consider the injection which induces
The map is injective since V is dense in . Let us show that it is surjective. Suppose so that is regular. Write
By example 6, we have and . Since are all dense in V we have injections and so that . It is easy to show that this means .
Hence induces an isomorphism of the coordinate rings . If V is affine, by proposition 1 here would be an isomorphism of varieties, which is a contradiction since is not surjective.