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.



Extending birational isomorphisms between planar curves to the P^2
In general the answer is **no**.


This kind of question is studied in more generality in the paper by Mella and Polastri "Equivalent birational embeddings", Bull Lond. Math. Soc. **41** (2009) 89-93, [http://arxiv.org/abs/0906.4858.](http://arxiv.org/abs/0906.4858)


They prove that two birational embeddings of $X$ in $\mathbf{P}^n$ are equivalent up to Cremona transformations of $\mathbf{P}^n$ as long as $n\geq \dim(X)+2$. For instance, any rational variety of codimension at least $2$ in $\mathbb{P}^n$ is Cremona equivalent to a linear space.


The case $n=2$ and $\dim(X)=1$ is outside this range, and indeed there are examples of birational plane curves that are not equivalent up to Cremona transformations, hence not equivalent under the action of $\textrm{Bir}(\mathbb{P}^2)$, since $\textrm{Bir}(\mathbb{P}^2)$ is generated by Cremona transformations.


The following example is given by Mella and Polastri in the last section of their paper. Take a general projection of a curve of bidegree $(1,d)$ on a quadric surface to $\mathbb{P}^2$. This is a plane rational curve $C$ of degree $d$ with only ordinary double points, hence there is a birational isomorphism $C \dashrightarrow L$, where $L$ is a line. However, one proves that if $d \geq 6$ then $C$ is **not** Cremona equivalent to $L$. 


The proof is based on the following 



> 
> **Lemma.** Let $X \subset \mathbb{P}^n$ be a rational variety of codimension $1$ and degree $d>1$. If $X$ is Cremona equivalent to a hyperplane, then the singularities of the pair $(\mathbb{P}^n, \frac{n+1}{d} X)$ are not canonical. 
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