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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!

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up vote 5 down vote accepted

If $\bar X$ is the smooth projective completion of $X$ (cf. Hartshorne, Chapter I, Corollary 6.10), you can consider $\phi$ as a rational map $\bar X-- \to Y$
But a rational map $\bar X--\to Y$ is actually a morphism $\bar{\phi}:\bar X\to Y$, because $\bar X$ is smooth and $Y$ is projective (cf. Hartshorne, Chapter I, Proposition 6.8).
Finally since $\bar{\phi}:\bar X\to Y$ is a birational morphism of smooth complete curves it is an isomorphism (cf. Hartshorne, Chapter I, Corollary 6.12).
And so the answer to your last question is that you can always view $X$ as an open subset of $Y$.

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