Today, we talk about the Segre Embedding. This will let us say a bit more about the products of varieties that were mentioned before.
We start out with the variety , which we can define as before with the universal property. We define a map by . We denote the image of this map by .
We’ll use to denote the homogeneous coordinates on . With these coordinates, is given as the zero set of the polynomials as vary. In fact, is an isomorphism of with .
The Segre embedding gives us a way of constructing products of varieties explicitly. Let and be quasi-projective varieties (so they could be affine, projective, open subsets of either, whatever). Then, they are contained in some projective spaces and . So then we can restrict the Segre embedding to , and we obtain a quasi-projective variety in . This is isomorphic to , and so we can take this to actually be . This demonstrates that products of quasi-projective varieties always exist and are quasi-projective.
Let’s look more closely at the simplest Segre variety. That is, . This is an embedding . The image is given by the equation , where were coordinates on the first factor and were from the second.
This shows that the image actually contains two families of lines. If we fix a point in either space, we get a linear map from , and so we get two collections of lines. Different lines in each collection are disjoint, and lines in different families intersect at a single point.
Now we want to look at the degree of . If we take a homogeneous polynomial of degree on , it will restrict, on , to a polynomial which is homogeneous in each collection of variables separately and of degree in each. So the Hilbert Polynomial is the product . This expands to . The degree is the lead coefficient multiplied by the degree factorial, and so is .
The Segre embedding also interacts nicely with the Veronese embedding. We can obtain all sorts of new varieties isomorphic to by combining the Veronese embedding and Segre embedding in various ways.
A nice fact about the two is that the diagonal has image under the Segre embedding in that is equal to if you choose the right subspace of .
That’s it for now, but we’ll return to Segre varieties later, if for no other reason because they tend to form nice special cases of other constructions.