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I see in this paper, page 46, the second sentence of 4.1, that every smooth variety over a characteristic $0$ field can be embedded into a proper smooth variety with normal crossing boundary, and the reason is Hironaka's resolution of singularity and Nagata's paper (which says every variety embedds in to a proper variety as an open set).

But I can't see how (maybe I have left something), since Hironaka's paper only shows there's a birational morphism with a proper smooth variety, but has nothing to do with embedding and information about the boundary, and I don't know how to combine the results of these two papers..

Any idea or better references are welcome.

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    $\begingroup$ Before asking here, you could maybe have a look at Hironaka's paper? This is Corollary 3, §5 in the Annals paper. $\endgroup$
    – abx
    Commented Nov 27 at 5:06

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