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Suppose $K$ is an infinite field of char. $0$. Let $C$ be an $K$-irreducible affine algebraic $K$-curve and suppose that $P_1,\ldots, P_n$ are non-singular $K$-points of $C$. Can one always find a plane curve $D$ and a birational map (defined over $K$) $f:C\dashrightarrow D$ such that for all $1\leq i \leq n$, $f(P_i)$ is defined and is a non-singular $K$-point of $D$?

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    $\begingroup$ This is possible and follows from the general yoga of projections. $\endgroup$ – Mohan Aug 22 '18 at 17:40

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