15
$\begingroup$

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

$\endgroup$
  • 9
    $\begingroup$ 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
18
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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
  • 6
    $\begingroup$ 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
  • 1
    $\begingroup$ For an english reference, see: stacks.math.columbia.edu/tag/0BCU $\endgroup$ – Asvin Jul 26 '18 at 23:36
2
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.