# Varieties II: Quasiprojective Varieties

This post contains the definition of (quasi-) projective varieties, regular functions, morphisms, projective coordinate ring, the Nullstellensatz. It also contains the global section of sheaf of regular function over distinguished open sets and the fact that affine neighborhoods form a basis for the Zariski topology

**Projective varieties**

Let be an algebraically closed field. We want to parametrize the 1-dimensional subspaces of . In coordinates, we take away the origin and identify two points and if there exists such that . The resulting space is called the projective space, denoted by or just . The coordinates are denoted as , called the homogeneous coordinates.

Again we want to consider polynomials as functions. Contrary to the affine case, polynomials aren’t really functions on a projective space, because of the scalar factor. However, if we consider only the homogeneous polynomials (i.e. those with each monomial having the same degree)

we can define the zero set.

Definition 1A projective variety is a subset of , that is the common zeros of a set of homogeneous polynomials.

**Remark**

- The set of polynomials can be taken to be finite. (Hilbert Basis Theorem)

**Some terminologies**

1. **the Zariski topology**

Same as the affine case – consider the topology on with the closed sets being projective varieties. It is easy to verify that this is a topology:

- ,
- Union of two closed sets:
- Intersection of closed sets: just put all the equations together.

2. **the ideal of X **

This is the ideal generated by all the homogeneous polynomials in that vanish on . (Remember that in general, a polynomial as a function in projective space does not make sense)

It is immediate that is a graded ideal/homogeneous ideal, i.e. for each , each of its homogeneous components also lies in .

3. **the projective coordinate ring **

Consider as the coordinates of the projective space , and restrict it on . If we allow free multiplication and addition, this is what we will get.

**The Nullstellensatz**

Similar to the affine case, we want to develop a correspondence such that:

- for a subset , we have the homogeneous ideal generated by all the homogeneous polynomials vanishing on .
- for a homogeneous ideal , take a generating set of homogeneous polynomials, and call their common zeros .

Similarly, we have

- If , then
- If are homogeneous ideals of , then
- If is a homogeneous ideal of ,
- If , then (closure in Zariski topology)

Again, we want to find a Nullstellensatz such that the third condition is an equality. However, this time it CANNOT be as nice as the affine case, for such an equality would imply a one-one correspondence between projective varieties of and radical homogeneous ideals of , which is false. (e.g. common zero of is empty, but this ideal is not the entire ring)

This counterexample is actually the worse we can get, as we shall see.

Theorem 1 (Hilbert’s Nullstellensatz, projective case)Let be a homogeneous ideal, where is algebraically closed field. If , then .

**Proof**

It suffices to show that .

Let be a homogeneous polynomial such that for some , vanish on . If we are done. Suppose not, then we see that cannot be a constant since . So is homogeneous of degree at least 1, which means that it would vanish at when regarded as a function on . Regard as an usual ideal in and apply the usual Nullstellensatz, we are done.

What if ? Regarding as an ideal of and as a subset of , we see that or .

So any homogeneous ideal that contains a power of would have empty common zero set in projective space. As in the affine case, it seems more natural to correspond and . So is called the **irrelevant maximal ideal**, and if we ignore all the radical ideals that contain , we have

Corollary 1Let be algebraically closed. The association is a 1-1 correspondence betweenprojective subvarieties ofandradical homogeneous ideals ofthat do not contain . Moreover, is irreducible iff is prime.

**Affine cover of projective varieties**

The projective space has a cover by affine space: for the piece , every point corresponds to in . In the other direction, a point corresponds to where 1 is inserted to the -th position.

We can also carry this cover to a projective subvariety. Let be an irreducible projective subvariety. Then is closed in , which is like an affine space. If is an affine variety, how would its defining equations be related to that of ?

Let , where each are homogeneous polynomials. is then defined as

where 1 is inserted to the -th position. This is exactly the process of **dehomogenenization.** It is then clear that is exactly the dehomogenization of .

Notice that if is irreducible, then the closure of in is exactly . (More generally, an open subset of an irreducible set is dense. If not, is a nontrivial decomposition) So we can also work on the “inverse problem”: Given an irreducible affine variety , embed it into (say, by identifying the affine space with ), then how would the defining equations of be related to that of ?

We can guess the answer – it should be the inverse process of dehomogenization. Given a polynomial in , we can of course insert the powers of in each term such that the polynomial becomes homogeneous. This is called **homogenization**.

Proposition 1Given an irreducible affine variety, and embed it into via . Then is the ideal generated by the homogenization of each term of .

**Proof**

Denote the ideal generated by the homogenization of each term of as . We want to show that it is exactly . Note that is generated by homogeneous polynomials, so it is a homogeneous ideal.

Clearly, , so , which implies contains .

For the other direction, consider . If we dehomogenize this polynomial by substituting by 1, the new polynomial . Then when we homogenize back, , implying that , showing the equality.

So far we talked about projective varieties and affine varieties. Just now we have seen that affine varieties can be embedded into the projective space and is a **locally closed **subset of . (meaning that it is an intersection of closed subset and open subset) This is the class of varieties we would focus on.

Definition 2Aquasi-projectivevariety in is an intersection of a closed subset and an open subset in Zariski topology.

From now on a variety would mean a quasi-projective variety.

**Regular functions of quasi-projective varieties**

In the affine variety case, we first looked at the quotient of polynomials (with the denominator not vanishing anywhere on the variety), and proved that it’s the same as polynomial function.

In the projective case, our analogue of polynomial is homogeneous polynomials. However, when we want to do quotient, we would want the numerator and the denominator to have the same degree – otherwise the function is still not well defined. So our first attempt is, for a quasi-projective variety , a regular function is a quotient of homogeneous polynomials of the same degree with denominator nonvanishing on . This, however, has some deficiencies.

*Motivating case*

Consider minus the point . The function defines a regular map from minus , for the denominator vanishes when , i.e. ([0,1,1]) or ([0,1,-1]).

has another expression on , that is . It is unnatural to distinguish these two as functions, while their denominators do vanish at different places. In particular, the expression “extends” the definition to the point , so should make sense as a function on , even though the value at is not initially defined.

This suggests the following definition,

Definition 3Let be a variety. A regular function from to is a map such that for each , there exists a neighborhood around and homogeneous polynomials of the same degree such that is nonvanishing on and on .

We can show that this is consistent with our definition for the affine case. In fact,

Proposition 2

- For ( is a polynomial, and an affine variety), a regular map is of the form
- For ( is a homogeneous polynomial, and is a projective variety), a regular map is of the form , where the numerator has the same degree as the denominator.

**Proof**

We will prove the first statement only as these statements are analogous.

First notice that the sets (as varies) actually form a basis of the Zariski topology on $X$. (For this reason they are called the **distinguished open sets**)

*A better local representation of regular function*

Consider an arbitrary regular function . For each , there exists a neighborhood such that on . Since form a basis, we can assume that by shrinking if needed.

does not vanish at all on , meaning that . Taking on both sides, , i.e. For some , on . Therefore the regular function has another local expression

Notice furthermore that . So let us replace by . That means that for any , there exists a neighorhood such that on .

*Putting the presentations together*

Notice that is compact, for if where are polynomials and is an index set, then , where is the ideal generated by all . Applying on both sides, , i.e. there exists , . Say , where , as functions on . Then , showing compactness.

So suppose that is covered by , where are chosen as before. As before, , so there exists polynomials such that

This tells us how to patch the local functions together. They are supposed to be same fraction except that they are defined on different domains, so these functions should all be the same as

which finishes the proof.

**Remarks**

- For the projective case, is also a basis.
- The regular functions for a variety clearly form a ring, denoted as .

Corollary 2

- For an affine variety, a regular function is a polynomial map.
- For an irreducible projective variety, a regular function is constant.

**Proof**

Take .

Proposition 3Let be an irreducible variety and be two regular maps. If they agree on an open set, then they are the same.

**Proof**

The set is closed because is regular, and it contains an open set, which must be dense in . So .

**Maps between quasi-projective varieties**

The definition of regular functions suggests a local definition for regular maps as well, so let us define

Definition 4Let be varieties. For a map ,

- If is quasi-affine (intersection open and closed set in ), is regular if each component function is regular.
- If is quasi-projective, consider the affine cover of . is regular if the restrictions of to is regular.

There is another definition of regular maps, that makes use of the regular functions and is analogous to differentiable functions.

Proposition 4Let be varieties. A map is regular if and only if is continuous and for any regular function ( open in ), the composition is also regular.

**Proof**

We first show that if is quasi-affine, a regular map is continuous.

Let be a closed subset of , and we want to show that is closed. Consider any . We will show that there exists a neighborhood such that is closed in , then we are done by this lemma.

Lemma 1Let be a topological space with an open cover . A subset is closed iff is closed in for all .

Let has coordinate functions . We can pick a neighborhood such that all these are of the form , where are homogeneous of the same degree. If is the common zeros of polynomials , then is the common zeros of these compose with $latex\displaystyle \frac{P_i}{Q_i}$. This is closed once we clear the denominators.

For the regular function part, take any regular function . We have an open cover of such that each is quasi-affine, and by definition it is clear that is regular if and only if each restriction to is regular. Thus we may assume that is quasi-affine by shrinking if needed. The regularity of can be shown using a similar argument of closedness above.

If is quasi-affine, note that the projection maps are regular functions, so is also regular by the hypothesis.

For quasi-projective case, we are done by the following lemma,

Lemma 2Let be varieties, and be a continuous map such that for any regular function ( is open in ), is also regular. If is open and is open such that , then the restriction of on also satisfies the fore mentioned property.

**Proof of Lemma 2
**

The key is that the restriction of a regular function on an open subset is still regular, which is clear.

Corollary 3The composition of regular maps is regular.

Knowing how to define a regular map, we now have the notion of isomorphism of varieties. From now on a variety that is isomorphic to an affine variety will be called **affine**. Similarly a variety that is isomorphic to a projective variety will be called projective.

**Regular maps and ring of regular functions**

A regular map induces a -algebra homomorphism .

Can we generalize Proposition 2 in the last post? Examining the proof, we see that we used the coordinates of . For there, we only regard it as regular functions on . Thus the proof generalizes to give

Proposition 5For two varieties , where is affine, we have anaturalisomorphism

**Projective coordinate ring and ring of regular functions
**

For affine varieties we have seen that . However for projective coordinate ring, this is not the case. For example in proposition 2, we see that , while . This shows that affine varieties are quite special.

In general, the ring of regular functions could be wild. It is mentioned in Shafarevich’s book that Rees and Nagata constructed examples of quasiprojective varieties such that is not finitely generated, though I can’t find these examples anywhere.

**Distinguished open sets**

We used this basis in the proof of proposition 2. Since the ring of regular functions on the basis is nice (as shown in the proposition), these sets are important.

Proposition 6Every point of a variety has an affine neighborhood.

**Proof**

If is quasi-projective, it has an affine open cover, so WLOG assume that is quasi-affine. As we have seen that is a basis, we would be done if we can show that is affine.

Let , where is affine. Suppose that is the common zeros of . Then notice that is isomorphic to the set defined by , , which is affine.

dude, where do u go to school ? Cuhk ? or HKU ?

I study at UST.