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Does anybody know a textbook or journal reference which contains the proof of the "well-known" fact that a smooth projective curve minus a point (or even finitely many points) is affine? The proof appears here on math.SE multiple times, in Ravi Vakil's notes (but not in his book, at far as I can tell) and other discussion forums. I need something citable. The case of elliptic curves would suffice.

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HeinrichD
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    I thought it was in Görtz/Wedhorn, §15.5, but it's not. You can always cite Hartshorne, Exercise IV.1.4, but I suppose that's not ideal… – Takumi Murayama Oct 20 '16 at 15:43
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    A point on a curve is an ample divisor, and the complement of an ample divisor is always affine. – Sasha Oct 20 '16 at 17:05
  • @Sasha: I am looking for a reference which is citable. – HeinrichD Oct 20 '16 at 17:26
  • @TakumiMurayama: Thank you. This is in fact not ideal, because it is an exercise. – HeinrichD Oct 20 '16 at 17:28
  • Seems relevant: http://math.stackexchange.com/questions/1594240/must-a-proper-curve-minus-a-point-be-affine?noredirect=1&lq=1

    (note that projective $\Rightarrow$ proper)

     – oxeimon Oct 21 '16 at 04:36

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This is practically Sasha's proof, but you can cite Proposition 5 in:

Goodman, Jacob Eli. "Affine open subsets of algebraic varieties and ample divisors." Ann. of Math. (2) 89 (1969), pp. 160–183. DOI: 10.2307/1970814. MR: 0242843.

Takumi Murayama
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  • Thank you. But why should a curve minus a point be proper? Prop. 5 assumes a proper open subscheme. – HeinrichD Oct 21 '16 at 07:20
  • @HeinrichD By "proper open" Goodman means that the open set in question is not equal to the whole curve. – Takumi Murayama Oct 21 '16 at 07:38
  • Ah, I was mixing up the two meanings of "proper". Thank you for clearing this up. I should have noticed this because the paper uses the term "complete". – HeinrichD Oct 21 '16 at 07:44