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!