# 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.

Lemma 1.Let be a graded ring; a proper homogeneous ideal is prime if and only if:

.

**Proof**

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. ♦

**Exercise A**

Prove that a proper homogeneous ideal of a graded ring *A* is primary if and only if

.

Proposition 1.Suppose the closed subset corresponds to the homogeneous radical ideal , , where .

Then V is irreducible if and only if is prime.

**Proof**

(⇒) 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 . ♦

Corollary 1.Let be a non-empty closed subset. Then is irreducible if and only if is irreducible.

**Proof**

By proposition 1, *V* is irreducible if and only if is prime. But (from an exercise here) so the result follows. ♦

# Quasi-projective Varieties

Recall that a projective variety is a closed subset of some .

Definition.A

quasi-projectivevariety 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.

Definition.Let and be quasi-projective varieties and be a function.

We say is

regularat if there is an open neighbourhood U of in V such thatfor some homogeneous of the same degree.

We say is

regularif it is regular at every , in which case we also say is amorphism of quasi-projective varieties.

From the above definitions, we obtain the category of all quasi-projective varieties and morphisms between them.

### Example 1

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:

Lemma 2.Let be regular under the new definition. Then there exist polynomials which represent .

**Proof**

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 . ♦

### Example 2

Take the map given by

Note that the same set of polynomials works globally over the whole of .

### Example 3

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 .

# Isomorphisms

Definition.Consider the category of all quasi-projective k-varieties, with morphisms defined as above. Two such varieties are said to be

isomorphicif they are isomorphic in the category.A quasi-projective variety is said to be

projectiveif it is isomorphic to a closed subset of some (this generalizes the existing definition of projective varieties);affineif it is isomorphic to an affine k-variety (closed subspace of some );quasi-affineif it is isomorphic to an open subset of an affine variety.

### Example 4

In example 3 above, we get an isomorphism since we have the reverse map

.

As an exercise, prove that and .

Definition.The

coordinate ringof a quasi-projective variety V is the set,

taken to be a k-algebra via point-wise addition and multiplication:

**Note**

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).

### Example 5

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**:

### Example 6

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:

### Example 7

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

where .

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.