Let $k$ be a field, assumed to have characteristic $0$ for simplicity (though this probably isn't necessary).

Let $A$ be a central simple algebra over $k$ of dimension $n^2$. Then the collection of left ideals of rank $n$ in $A$ may be given the structure of a variety $B(A)$, which turns out to be a Brauer-Severi variety (i.e. a form of projective space) of dimension $n-1$. Moreover a classical theorem states that every Brauer-Severi variety of dimension $n-1$ arises in this way.

Next recall that given two central simple algebras $A_1$ and $A_2$ over $k$, their tensor product $A_1 \otimes_k A_2$ is also a central simple algebra over $k$. My question concerns how the corresponding Brauer-Severi varieties are related.

How may one visualise $B(A_1 \otimes_k A_2)$ in terms of $B(A_1)$ and $B(A_2)$? In particular, is there a geometrical operation that one may perform on $B(A_1)$ and $B(A_2)$ to obtain $B(A_1 \otimes_k A_2)$?

This question is slightly vague; so let me give you an example of the kind of thing I want, but which unfortunately does not work. Namely, the only natural construction which I can think of here is the fibre product. But $B(A_1) \times B(A_2)$ is not isomorphic to $B(A_1 \otimes_k A_2)$ for two obvious reasons:

It has the wrong dimension.

It is isomorphic to a product of projective spaces, and not projective space, over an algebraic closure $\overline{k}$ of $k$.

`$B(A_1 \otimes_k A_2)$`

corresponds to the form of projective space the product`$B(A_1) \times B(A_2)$`

Segre embeds into. – Michael Stoll Apr 8 '13 at 20:21