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 \mathbb{P}^n\times\mathbb{P}^m, which we can define as before with the universal property. We define a map \sigma:\mathbb{P}^n\times\mathbb{P}^m\to \mathbb{P}^{(m+1)(n+1)-1} by ((x_0:\ldots:x_n),(y_0:\ldots:y_m))\mapsto (\ldots:x_iy_j:\dots). We denote the image of this map by \Sigma_{n,m}.

We’ll use z_{ij} to denote the homogeneous coordinates on \mathbb{P}^{(m+1)(n+1)-1}. With these coordinates, \Sigma_{n,m} is given as the zero set of the polynomials z_{ab}z_{cd}-z_{cb}z_{ad} as a,b,c,d vary. In fact, \sigma is an isomorphism of \mathbb{P}^n\times\mathbb{P}^m with \Sigma_{n,m}.

The Segre embedding gives us a way of constructing products of varieties explicitly. Let V and W be quasi-projective varieties (so they could be affine, projective, open subsets of either, whatever). Then, they are contained in some projective spaces V\subset\mathbb{P}^n and W\subset\mathbb{P}^m. So then we can restrict the Segre embedding to V\times W, and we obtain a quasi-projective variety in \mathbb{P}^{(m+1)(n+1)-1}. This is isomorphic to V\times W, and so we can take this to actually be V\times W. 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, n=m=1. This is an embedding \mathbb{P}^1\times\mathbb{P}^1\to \mathbb{P}^3. The image is given by the equation xw-yz=0, where (x:y) were coordinates on the first factor and (z:w) 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 \mathbb{P}^1\to\mathbb{P}^3, 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 \Sigma_{n,m}. If we take a homogeneous polynomial of degree \ell on \mathbb{P}^{(m+1)(n+1)-1}, it will restrict, on \Sigma_{n,m}, to a polynomial which is homogeneous in each collection of variables separately and of degree \ell in each. So the Hilbert Polynomial is the product \binom{m+\ell}{m}\binom{n+\ell}{n}. This expands to \frac{1}{m!n!}\ell^{m+n}+\ldots. The degree is the lead coefficient multiplied by the degree factorial, and so is \frac{(m+n)!}{m!n!}=\binom{m+n}{n}.

The Segre embedding also interacts nicely with the Veronese embedding. We can obtain all sorts of new varieties isomorphic to \mathbb{P}^n\times\mathbb{P}^m by combining the Veronese embedding and Segre embedding in various ways.

A nice fact about the two is that the diagonal \Delta\subset \mathbb{P}^n\times\mathbb{P}^n has image under the Segre embedding in \mathbb{P}^{n^2+2n} that is equal to V_{2,n} if you choose the right subspace of \mathbb{P}^{n^2+2n}.

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.