TITLE: Extending birational isomorphisms between planar curves to the P^2
QUESTION [10 upvotes]: Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a birational isomorphism $F:\mathbb{P}^2_k \to \mathbb{P}^2_k$ such that $F$ and $f$ agree on a (non empty) open subset of $C$?
I am mainly interested in the case when $k=\mathbb Q$. 
[Edit] Since it is apparently easy to read over I will state it here explicitly. The map $F$ is allowed to be a birational isomorphism. It is clear that my statement is false when you want $F$ to be an isomorphism since it has to send curves of the same degree to curves of the same degree.
If we take for example $C$ to be the curve $x=0$ and $D$ to be the curve $y^2z=x^3+x^2z$ then $C$ and $D$ are birationally equivalent but clearly no automorphism of $\mathbb{P}^2_k$ sends C to D. However it is possible with a birational isomorphism.

REPLY [4 votes]: Another way to see that a nodal rational curve of degree $\ge 6$ cannot be sent to a line by a birational map of the plane is to see that it cannot be contracted by a birational map of the plane. Each birational map of the plane decomposes into blow-ups and contractions of smooth $(-1)$-curves (curves isomorphic to $\mathbb{P}^1$ and of self-intersection $-1$). If you want to contract the curve, you have to blow-up all nodes. The number of nodes is $(d-2)\cdot (d-1)/2$, and the self-intersection of the curve on $\mathbb{P}^2$ is $d^2$. After blowing-up, the self-intersection decreases by $4$ for each double point, so becomes $d^2-2\cdot (d-1)(d-2)=6d-d^2+4$. If $d\ge 6$, this number is $\le -2$, so the curve is not contractible. If $d\le 5$, the curve is contractible and in fact one can check easily that it can be sent onto a line by a birational map of the plane.
Similarly, one sees that a nodal rational curve of degree $d\ge 6$ cannot be sent onto a nodal rational curve of degree $d'\ge 6$ when $d\not=d'$.
Similar arguments work with curves of genus $1$. See for example  Proposition 3.3.3 of "On birational transformations of pairs in the complex plane", J. Blanc, I. Pan, T. Vust, Geom. Dedicata 139 (2009), 57-73.