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Let $C$ be a smooth curve with function field $K$. The projective line $\mathbf{P}^1_C$ is a smooth model over $C$ for the projective line $\mathbf{P}^1_K$. Does there exist another model of $\mathbf{P}^1_K$, say $X$, and a surjective birational morphism $\mathbf{P}^1_{C} \to X$?

On the generic fibre this morphism is the identity.

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    Dear Haki, Do you know any examples of singular rational curves, e.g. as alluded to in QiL's answer below? Regards,2011-12-16

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