Let $X$ be a connected affine variety over an algebraically closed field $k$, and let $X \subset Y$ be a compactification, by which I mean $Y$ is a proper variety (or projective if you prefer), and $X$ is embedded as an open dense subset.

I am guessing that it is not always the case that $Y\setminus X$ is a divisor, one could imagine it being a single point with a horrible singularity. But if $Y$ is smooth or even normal, is it the case that $Y\setminus X$ is always a divisor? Does anybody know a proof of such a result?

Thanks, Dan

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    $\begingroup$ Affirmative even in the normal case, but this is a good exercise to work on more on your own. Hint: look at local ring at a generic point of the complement (assuming it is non-empty). As an aside, this plays a key role in producing the ample divisor in the proof that abelian varieties are projective! $\endgroup$ – BCnrd Apr 21 '10 at 12:24
  • $\begingroup$ A side remark: such a divisor must also be connected if dimension of $X$ is $\geq{2}$ ! $\endgroup$ – Maharana May 2 '10 at 7:02

It it true for any $Y$: see Corollaire 21.12.7 of EGAIV.

  • $\begingroup$ Thanks, although unfortuantly I cant speak french... Ill have a go and try to prove it myself. $\endgroup$ – Daniel Loughran Apr 21 '10 at 16:16
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    $\begingroup$ Have fun. By considering the normalization of $Y$, it is easy to reduce to the normal case. You don't need to speak French to read EGA. You don't even to really read it, only learn a few technical words and some funny-looking verbal forms. $\endgroup$ – Angelo Apr 21 '10 at 19:20
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    $\begingroup$ For an english reference, see: stacks.math.columbia.edu/tag/0BCU $\endgroup$ – Asvin Jul 26 '18 at 23:36

Goodman wrote a paper on a related subject entitled: "Affine open subsets of algebraic varieties and ample divisors". You might find something in there that's relevant.

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    $\begingroup$ I think this paper discusses the question: If the complement of a divisor on a projective variety is affine, what positivity properties does the divisor have? In the OPs question, $X$ does not have to be projective. $\endgroup$ – J.C. Ottem Apr 21 '11 at 8:56

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