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.

  • 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

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.

  • $\begingroup$ 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

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