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.