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 .
The triplet is a product in the category of quasi-projective varieties.
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.
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 .
Prove that is the closed subspace of defined by
over all .
Products of Quasi-Projective Varieties
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.
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.
The product of two projective (resp. quasi-projective) varieties exists and is projective (resp. quasi-projective).
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.
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 . ♦
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 .
For any point in a quasi-projective variety V, there is an open neighbourhood U, , which is affine.
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 . ♦
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.
if V and W are quasi-projective varieties, then
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.
Let be a non-empty closed subset. Then
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 . ♦