It is a well-known fact that if $X$ is a projective curve and $p \in X$ a smooth point, then any rational map $X \to Y$, $Y$ a projective variety, extends to a rational map $X \to Y$ regular at $p$. This is proposition I.6.8 in Hartshorne (in the case of $X$ an abstract non-singular curve), for example. However, the two proofs I have seen both assume that it suffices to consider the case $Y = \mathbb{P}^n$. As I understand it, this is because morphisms of projective varieties are proper, and in particular the image is closed. Where I can find a proof of this, in the case of varieties only? I found a proof here by Akhil Mathew, but I got lost when he started talking about base change.
Why are projective morphisms closed?
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algebraic-geometry
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0Does it help when I tell you that base change is simply another word for [pull-back](http://en.wikipedia.org/wiki/Fibred_product)? In Wikipedia's notation $p_2$ is the base change of $f$ along $g$. (Think of $X$ as a bundle with base $Z$ and $P = X \times_Z Y$ as a bundle with base $Y$ and $f$ and $p_2$ the respective bundle projections). – 2011-06-07
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3Dear Zhen, A projective variety, by definition, is something that is closed in projective space. So if you prove that a rational map $X \dashrightarrow Y$ extends to a map $X \to \mathbb{P}^n$, then the image must lie inside $Y$ (because $Y$ is closed). Now since $X$ is integral this means it scheme-theoretically factors through $Y$ as well. – 2011-06-07
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1Dear @Akhil: Thanks for pointing the obvious out. My point set topology is too rusty to tell when intuition from metric spaces applies in non-metrisable contexts... – 2011-06-07
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0The "well-known fact" you mention in your first sentence is surely misstated: there is no relation betweemn $p$ and the rational map... When quoting things from Hartshorne, it is good to include the chapter number (each chapter has its own Proposition 6.8) – 2011-06-07
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0You probably really mean "rational map regular at $p$" instead of "morphism" at the end of the 1st sentence, no? – 2011-06-07
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0@Marinano: Of course. This carelessness is probably going to cost me in my exams... – 2011-06-07