Can a birational map be completed to a proper map? Let $f\colon X\to Y$ be a birational map of complex algebraic varieties. Are there necessarily open immersions of $X$ and $Y$ into varieties $X’$ and $Y’$, resp., which admit a proper morphism $f’\colon X’\to Y’$ extending $f$?
 A: This follows from Nagata's compactification theorem [Nag62] and its relative version [Nag63]. Indeed, one may compactify $Y$ to get an open immersion $Y \hookrightarrow Y'$ with $Y'$ proper [Nag62]. Replacing $Y'$ by the reduced structure on the closure of $Y$, we may assume $Y'$ is integral.
Now apply relative compactification to $X \to Y'$ to get an open immersion $X \hookrightarrow X'$ of $Y'$-schemes [Nag63]. Again, we may assume $X'$ is integral. This gives a commutative diagram
$$\begin{array}{ccc}X & \stackrel{f}\longrightarrow & Y \\ \downarrow & & \downarrow \\ X' & \stackrel{f'}\longrightarrow & Y',\!\!\!\end{array}$$
and we have $f'^{-1}(Y) = X$ if and only if $f$ is proper. Indeed, if $f'^{-1}(Y) = X$ then $f = f' \times_{Y'} Y$ is proper. Conversely, if $f$ is proper, then $X \to f'^{-1}(Y)$ is an open immersion of proper $Y$-schemes, hence a closed immersion. It is also dense since $X \subseteq X'$ is dense, so we must have $X = f'^{-1}(Y)$. $\square$
This appears as Lemma 5.1 in a preprint of mine [vDdB], but surely must be written up somewhere else as well. (Does anybody know a canonical reference?)
References.
[vDdB] R. van Dobben de Bruyn, The equivalence of several conjectures on independence of $\ell$. arXiv: 1808.00119
[Nag62] M. Nagata, Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ. 2, p. 1-10 (1962). ZBL0109.39503.
[Nag63] M. Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety. J. Math. Kyoto Univ. 3, p. 89-102 (1963). ZBL0223.14011.
