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Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\setminus X $ and $\bar Y\setminus Y$ are normal crossing divisors, and $f$ extends to $\bar X\to \bar Y$? I'm not familiar with Hironaka's resolution of singularity, and any idea and references are welcome.

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    $\begingroup$ To be fair, most working algebraic geometers' knowledge of resolution of singularities consists of knowing the statements (and the proof in dimension $2$). You can deduce a positive answer to your question easily from Hironaka (plus Nagata compactification). Choose compactifications $\bar X$ and $\bar Y$ of $X$ and $Y$ respectively. Blowing up in $\bar Y \setminus Y$, you can make $\bar Y$ smooth with normal crossings boundary. Replace $\bar X$ by the closure of the graph of $f \colon X \to \bar Y$ in $\bar X \times \bar Y$ to assume $f$ extends, and then apply resolution to $\bar X$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 27 at 12:38

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