TITLE: Curve over ring, covered by two affines
QUESTION [8 upvotes]: The reference is http://www.math.columbia.edu/~masdeu/files/notes/FallSeminar.pdf, page 9:

Let now $C/R$ be a curve over a noetherian ring $R$; this means that $C$ is smooth, connected, integral, proper and of relative dimension $1$ over $R$. For curves it is always possible to find a covering by two open affines, say $\{U,V\}$.

To me this is not so clear. How would one find $U$ and $V$? If $C$ was a projective curve over a field $K$, given by $f(X,Y,Z) = 0$, I think we can find $U$ and $V$ by dehomogenizing with respect to $Z$ or $Y$, respectively. If $R$ is not a field, I'm not sure it makes sense to dehomogenize - basically 'dividing by $Z$' could be a problem. Can we still do this?

REPLY [2 votes]: I think this is not so easy, even when $C$ is projective over $R$. 
The answer is yes if $R$ is a local ring. In general, even when $R$ is a Dedekind ring, I don't know whether this is true. There are some related discussions at https://mathoverflow.net/questions/112085/.