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Let $k$ be a field and $\phi: k^*\rightarrow X$ be a morphism to a variety $X$.

My questions are:

1.When $X$ quasiprojective, why $\phi$ can be extended to $\bar\phi:\mathbb P^1\rightarrow \overline X$ where $\overline X$ is the closure of $X$ in $\mathbb P^n$?

2.Why $\overline{\phi(k^*)}-\phi(k^*)$ must be equal to $lim_{x\rightarrow 0}\phi(x)$ or $lim_{x\rightarrow\infty}\phi(x)$? (I'm also not really sure where do we take the closure for $\phi(k^*)$. in $X$ or in $\mathbb P^n$?)

3.Why $\phi$ is proper if and only if these limits do not exist in $X$?

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Hydrogen
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  • I think there might be a generalization. Let $X$ be scheme and $Y$ be quasiprojective scheme $\phi: U\rightarrow Y$ be a morphism from a dense open set. Then $\phi$ can be extended to $\overline{\phi}:X\rightarrow \overline Y$. Is this statement true? If it is true, is the extension unique or does that depend on the immersion $Y\subseteq \mathbb P^n$? – Hydrogen Jan 22 '21 at 22:41
  • The first question is addressed in variety-theoretic language here, and in scheme-theoretic language here. The generalization in your comment is false, should probably be edited in to your question if you're interested in an answer, and has been addressed on this site before here. You may find it helpful to do a bit more searching on your own next time. – KReiser Jan 22 '21 at 22:57

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