# Segre Embedding

Throughout this article, *k* is a fixed algebraically closed field. We wish to construct the product in the category of quasi-projective varieties.

For our first example, let be the *projective* variety defined by the homogeneous equation . We define maps as follows

Note that the maps are well-defined: if then since we have .

Proposition 1.The triplet is a product in the category of quasi-projective varieties.

**Proof**

Let be a quasi-projective variety and be morphisms. We will define the corresponding as follows. For each , there is an open neighbourhood *U* of **w** such that and where are homogeneous of the same degree and either or has no zero in *U*. Same holds for .

Now define by . Clearly the image of *f* lies in *V* so we get a morphism . It is easy to see that and . Repeating this construction over an open cover of *W*, we obtain our desired . ♦

Using similar techniques, we can show the following.

Proposition 2.For any , the product exists in the category of quasi-projective varieties and is a projective variety.

Specifically, the product is the image of the **Segre embedding**

,

where the projective coordinates of are indxed by with and .

We denote the image of this map by .

**Exercise A**

Prove that is the closed subspace of defined by

over all .

# Products of Quasi-Projective Varieties

Proposition 3.If and are open (resp. closed), so is the image of in .

In particular, the topology on is at least as fine as the product topology.

**Proof**

It suffices to prove the case where and are open. Pick any and ; without loss of generality say .

Since is open in there exists a homogeneous such that . Similarly, there exists a homogeneous such that . Then

so the image of in *V* is open. ♦

As in the product of affine varieties, the topology on is in general strictly finer than the product topology. This is already clear in the case *m* = *n* = 1, since has the cofinite topology.

Corollary 1.The product of two projective (resp. quasi-projective) varieties exists and is projective (resp. quasi-projective).

**Note**

In the following proof, we say a subset of a topological space is **locally closed** if it is an intersection of an open subset and a closed subset. Thus every quasi-projective variety (resp. quasi-affine variety) is a locally closed subspace of some (resp. ).

Prove the following properties as a simple exercise:

- an intersection of two locally closed subsets is locally closed;
- if
*Y*is a locally closed subset of*X*and*Z*is a locally closed subset of*Y*then*Z*is a locally closed subset of*X*; - a subset
*Y*of*X*is locally closed if and only if*Y*is open in its closure in*X*.

**Proof**

If and are closed (resp. locally closed), so is the image *W* of in by proposition 3. The projections and then restrict to and .

Let us show that is the product of and in the category of quasi-projective varieties.

If *X* is any quasi-projective variety and , are any morphisms then and induce ; the image of *f* lies in *W* so we obtain an induced . ♦

**Exercise B**

1. Let be the set of points satisfying . Find a set of homogeneous polynomials in which define the image of *W*.

2. More generally prove that a subset is closed if and only if its corresponding subset is the set of solutions of some *bihomogeneous polynomials*

,

i.e. *F* is homogeneous as a polynomial in as well as .

# Dimensions

Lemma 1.For any point in a quasi-projective variety V, there is an open neighbourhood U, , which is affine.

**Proof**

Suppose is a locally closed subset. Without loss of generality, so is contained in , a locally closed subset of . Now *W* is open in , its closure in . By an analogue of proposition 1 here, we can pick a basis of the topological space in the form of , where

Thus for some we have . Now we are done since is isomorphic to the affine variety with coordinate ring . ♦

**Exercise C**

Prove that if *V* and *W* are irreducible quasi-projective varieties, then is also irreducible. *Again, please be reminded that is not the product topology.*

Proposition 4.if

VandWare quasi-projective varieties, then.

**Proof**

Suppose *V* and *W* are irreducible; by lemma 1 we can pick open affine subsets and . Then is an open affine subset of the quasi-projective variety , which is irreducible by exercise C. Now

.

where the first and third equalities follow from proposition 3 here and the second is from proposition 2 here,

The general case is left as an exercise (write *V* and *W* as unions of irreducible components). ♦

Finally, we consider the dimension of the cone of a projective variety.

Proposition 5.Let be a non-empty closed subset. Then

.

**Proof**

Without loss of generality, suppose ; by proposition 3 here, and *V’* is a closed subset of . Also since cone(*V’*) is open in cone(*V*) we have . Now there is an isomorphism

Hence . ♦