In this article, we will consider algebraic geometry in the projective space. Throughout this article, *k* denotes an algebraically closed field.

# Projective Space

Definition.Let . On the set , we consider the equivalence relation:

The

projective n-spaceis the set of equivalence classes under this relation. An element of is denoted by for any representative .

For example .

The above only defines as a *set*; we need some geometric structure on it for the concept to be meaningful. Throughout this article we will fix , which is graded by degree.

Definition.Suppose is homogeneous and . We write if .

Note that the *value* is in general not well-defined. Indeed if we have two representatives for , then

Despite this, if and only if so the definition is sensible. As in the case of the affine *n*-space, we will define a correspondence between ideals of *B* and subsets of .

Definition.Let be a graded ideal. Write

.

For a finite sequence of homogeneous polynomials we also write for where .

Also let .

**Exercise A**

Prove the following, for any graded ideals and collection of graded ideals of *B*.

- .
- If is a set of homogeneous generators of , then

.

- .
- .

In the other direction, we define:

Definition.Let be any subset. Then denotes the (graded) ideal of B generated by:

.

In summary, we defined the following maps.

# Zariski Topology of Projective Space

We wish to define the Zariski topology on ; for that let us take subsets of which can be identified with the affine space . Fix ; let

Note that for the same point can be represented by where the *i*-th coordinate is 1. This gives a bijection . E.g. for *n* = 2, we have:

Note that for any , the intersection maps to an open subset of via both and . Indeed if *i* < *j* then is the set of all satisfying while is the set of all satisfying . Hence, the following is well-defined.

Definition.The

Zariski topologyon is defined by specifying every as an open subset, where obtains the Zariski topology of from .A

projective varietyis a closed subspace .

First, we have the following preliminary results.

Lemma 1.For any homogeneous , the set is (Zariski) closed in .

**Proof**

It suffices to show that is closed in for each . But , where . The same holds for . ♦

Definition.For any , let

be its

homogenization.

**Exercise B**

1. Prove that if , the homogenization of *fg* is the product of the homogenizations of *f* and *g*.

2. Let be the homogenization of a non-constant . Then *f* is irreducible if and only if *F* is irreducible. [ Hint: you may find lemma 2 here helpful. ]

The Zariski topology on is consistent with our earlier notions of closed subsets:

Proposition 1.A subset is closed under the Zariski topology if and only if for some graded ideal .

**Proof**

(⇐) Suppose for some homogeneous . Since , by lemma 1 this is Zariski closed.

(⇒) Let ; it suffices to show that any is contained in for some homogeneous . Now is contained in some , say without loss of generality. Hence for some . If , then

, where *F* = homogenization of *f*. ♦

### Example

Let be the projective variety defined by the homogeneous equation . Then

- is cut out from by ;
- is cut out from by ;
- is cut out from by .

# Cone of Projective Variety

We wish to prove the bijective correspondence between graded radical ideals of *B* and closed subsets . For that, we can piggyback on existing results for the affine case.

Definition.Let be any subset. The

coneof C is.

**Note**

For any non-empty collection of subsets we have

Also, we have:

Lemma 2.If is a proper graded ideal then .

In particular, by proposition 1, the cone of a closed is closed in .

**Proof**

Note: if is *non-constant* homogeneous, then . Now pick a set of homogeneous generators for ; each is non-constant so

.

This completes the proof. ♦

Furthermore we have:

Lemma 3.A non-empty closed subset is of the form for some closed if and only if

.

When that happens, we call W a

closed conein .

**Proof**

(⇒) is obvious; for (⇐) clearly *W* = cone(*V*) for *some* subset . Let so that . It remains to show is graded, for we would get by lemma 2.

Indeed if , write as a sum of homogeneous components. Then for any and we have which gives

Thus vanish for any , i.e. . ♦

# Projective Nullstellensatz

Thus we have the following correspondences:

The top-left column is a bijection by lemma 3; the bottom row is a bijection by Nullstellensatz. In the proof of lemma 3, we also showed that for a closed cone , the ideal is graded. Conversely, if is graded, is the (non-empty) solution set of a collection of graded polynomials; hence it is a closed cone too.

Hence we have a bijection between

- closed subsets , and
- proper homogeneous radical ideals .

The correspondence takes and so

The last piece of the puzzle is the map which takes closed subsets of to homogeneous ideals of *B*. As an easy exercise, show that

.

However so we modify our bijection to:

Theorem (Projective Nullstellensatz).There is a bijection between:

- closed subsets ;
- homogeneous radical ideals such that where is the
irrelevant idealof B.

**Exercise C**

Prove that if is a homogeneous ideal then is empty if and only if contains a power of .

[ Hint: prove that the radical of 𝔞 is either (1) or *B*_{+}. ]