Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U \subset X$ containing the $x_i$?

To give some context, Lemma 4 of section 6.4 of "Neron models" of Bosch, Lutkebohmert, and Raynaud says that the answer should be 'yes' but I don't understand its proof (how does EGA IV, 8.10.5 apply to the $u_j$ constructed there?).

  • $\begingroup$ Are we allowed to use Raynaud-Gruson, Nagata compactification, etc., or is the concern that those theorems depend on this result? Probably there is an argument that uses none of those, but my first instinct is to reduce to the case that $X$ is a quasi-projective $R$-scheme. $\endgroup$ – Jason Starr Dec 8 '14 at 1:15
  • $\begingroup$ I'd be very happy to see any argument that works. For one thing, both Raynaud-Gruson and Nagata compactification have already been used earlier in the book (perhaps nonessentially for the eventual goal, but I didn't care to keep track), so I see no sin in using them again. $\endgroup$ – Question Mark Dec 8 '14 at 1:25
  • 1
    $\begingroup$ Okay, then you can replace $X$ by a projective modification that is an isomorphism over each of the $x_i$, and which itself admits a proper birational $R$-morphism to an $R$-projective scheme $Y$. Now replace $Y$ by its blowing up along the closures of the images of the points $x_i$. After a further projective modification of $X$, there is now a proper, birational morphism to $Y$ that is quasi-finite at every point $x_i$. Now replace $X$ by the Stein factorization of this morphism. This is projective. The projective case is straightforward. $\endgroup$ – Jason Starr Dec 8 '14 at 2:00
  • $\begingroup$ Thanks! Could you give more detail? That would be very helpful, since I don't have a lot of experience with the constructions you are using. For instance, doesn't your first sentence imply that $X$ is $R$-proper (how to arrange the map to $Y$ be proper?); to justify the first sentence, should I be using Chow's lemma stacks.math.columbia.edu/tag/088U together with stacks.math.columbia.edu/tag/080A and stacks.math.columbia.edu/tag/0807 ? Doesn't the latter imply that the blowing up in your second sentence is the identity, since the $x_i$ are still of height $1$? $\endgroup$ – Question Mark Dec 8 '14 at 5:55
  • 4
    $\begingroup$ Chow's lemma works directly. There is a proper surjective birational $p:X′\to X$ with $X′$ quasiprojective over $R$. Since $X$ is normal, $p$ is an isomorphism over an open subscheme $U\subset X$ containing each $x_i$. Clearly $U$ is quasiprojective, as a subscheme of $X'$. $\endgroup$ – Laurent Moret-Bailly Dec 8 '14 at 8:28

Your Answer

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

Browse other questions tagged or ask your own question.