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Suppose $X$ is a regular, Noetherian, separated, connected, one-dimensional scheme over a field $F$.

Questions: (1) Does there always exist a point $P\in A$ such that $X-P$ is an open affine subscheme in $X$ ?

(2)Assume $F$ is algebraic closed. Let $\eta$ be the unique generic point of $X$, and $K=k_{\eta}$ is the function field of $X$. Is the following statement right?

Statement: X is proper if and only if $\sum_{x\in X}ord_x(f)=0$ for all $f\in K$.

Remark: By this question, we know for a curve, proper is equivalent to projective.

Thanks in advance!

  • For (1), the basic idea is that by Riemann-Roch a sufficiently high multiple of $P$ is a very ample divisor, so in the associated projective embedding $X - P$ is a closed subscheme of an affine space (obtained by deleting the hyperplane that $P$ maps to), hence an affine variety. – Tabes Bridges Jul 02 '19 at 19:11
  • You have to be careful to assume that $X$ is also of finite type, I am not sure if part (2) is true without that assumption. – Pol van Hoften Jul 03 '19 at 18:13
  • @TabesBridges Thanks, I find this question. –  Jul 04 '19 at 02:18
  • @user45878 In this paper by Matthew Morrow, the author doesn't assume $X$ is of finite type when he define a curve in definition 2.6 on page 11. And by his definition, a curve is complete when the degree formula holds. So I wonder if completeness is equivalent to proper... –  Jul 04 '19 at 02:33

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