Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 6 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 $Y$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.