6
$\begingroup$

In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.

Could you give me a reference for this fact or help me to sketch a proof?

Thank you in advance.

$\endgroup$
1
  • 2
    $\begingroup$ You probably want to assume the curve is connected, otherwise you may end up with the disjoint union of an affinoid curve and a projective curve. $\endgroup$ – Jérôme Poineau Feb 25 '16 at 19:59
6
$\begingroup$

I think that the original reference is "Zariski's Main Theorem für affinoide Kurven" by K.-H. Fieseler (Mat. Ann. 251, 1980). He proves that a finite union of affinoid domains of a one-dimensional affinoid space is affinoid, but this is probably not enough to answer your question.

More generally, though, J. Fresnel and M. Matignon prove that a quasi-compact irreducible one-dimensional rigid space is either affinoid or projective ("Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique", Annali di Matematica Pura ed Applicata 145, 1986). You can also find a proof in chapter 6 of A. Ducros's book (see https://webusers.imj-prg.fr/~antoine.ducros/livre.html) in the language of Berkovich spaces.

$\endgroup$
2
  • $\begingroup$ Thank you for the reference. I can guess that Satz 2.1 proves what I am asking but unfortunately the proof is too long compared to my understanding of German. Is there another reference in either French or English? $\endgroup$ – Bear Feb 25 '16 at 20:39
  • $\begingroup$ Having a look at Fieseler's paper, it seems there is less than I thought there was. I corrected my answer and added other more complete references (in French). $\endgroup$ – Jérôme Poineau Feb 25 '16 at 21:01

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.