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Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes.

Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$?

Just to be clear: A projective curve is a smooth projective morphism $X\to S$ such that the geometric fibres are geometrically connected and of dimension 1.

In simple layman's terms: Can a non-projective variety contain a projective curve?

Feel free to replace "projective" by "proper". It probably won't change much.

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1 Answer 1

up vote 3 down vote accepted

Sure, $\mathbb{P}_k^2-\{pt\}$ is not proper over $Spec(k)$ and contains a proper $\mathbb{P}_k^1$.

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this is a closed immersion right? Any higher genus examples? –  Harry Apr 9 '12 at 18:35
    
Yes, john's example easily generalizes so that if you have any smooth projective curve $C$ embedded in some $\mathbb{P}^n$, you can always consider the closed immersion $C \hookrightarrow \mathbb{P}^n \setminus \{x\}$ for a point $x \in \mathbb{P}^n \setminus C$. –  Michael Joyce Apr 9 '12 at 18:39

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