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Let $X$ be a topological space equipped with maps into two spaces $\bar X_1$ and $\bar X_2$. Is there a standard notation/terminology for the closure $\bar X$ in $\bar X_1 \times \bar X_2$ of the diagonal map of $X$?

In my case $X$ is an affine algebraic surface (in fact just $\mathbb{C}^2$) which is isomorphic to Zariski open subsets of complete surfaces $\bar X_1$ and $\bar X_2$ and the maps $X \to \bar X_j$'s are the corresponding embeddings.

In a paper I wrote, I used the notation "birational join" for $\bar X$ following Spivakovsky, but the referee does not like it. (S)He suggested something like the "fiber product" $\bar X_1 \times_X \bar X_2$, but that would require the arrows $X \to \bar X_j$ to be reversed. Similarly the ``cofiber product'' requires the arrows $\bar X \to \bar X_j$ to be reversed.

Any suggestions would be much appreciated. Thanks!

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I believe that if you look at the category of varieties $Y$ with an open immersion $X \rightarrow Y$, then the "birational join" is the product (but not fiber product) in this category. This doesn't answer the question of what to call it, but it may explain the referee's comment. – Dustin Cartwright Feb 18 '12 at 1:28
Indeed. Thanks! – auniket Feb 18 '12 at 3:42
As a somewhat obvious answer, it is simply the closure of the universal map $X\to X\times X$ given by product topology. – B. Bischof Feb 18 '12 at 14:23

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