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.



Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)
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.


