I'm looking at algebraic curves at the moment (irreducible varieties of dimension 1). I have already had a great answer concerning the closed sets of a curve, so I'd thought I'd try my luck on stack exchange again; though I wouldn't be surprised to find out that this problem is quite a bit harder. Anyhoo, my question is this:
Let $X$ and $Y$ be non-singular affine and projective curves respectively, and let $\varphi:X\rightarrow Y$ be a birational map. I am wondering when $\varphi$ can be extended to an injective morphism $\widetilde{\varphi}:X\rightarrow Y$ In particular, when can $X$ be viewed as an open set of $Y$.
Thanks for any help!