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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 $P^2_k$. 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.

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.

Let $k$ be a field and let $C,D$ be two integral curves in $P^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $P^2_k$. To be precise, does there exist a birational isomorphism $F:P^2_k \to P^2_k$ 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 $P^2_k$ sends C to D. However it is possible with a birational isomorphism.

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 $P^2_k$. 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.

Let $k$ be a field and let $C,D$ be two integral curves in $P^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $P^2_k$. To be precise, does there exist a birational isomorphism $F:P^2_k \to P^2_k$ such that $F$ and $f$ agree on a (non empty) open subset of $C$?

I am mainly interested in the case when $k=\mathbb Q$.

Let $k$ be a field and let $C,D$ be two integral curves in $P^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $P^2_k$. To be precise, does there exist a birational isomorphism $F:P^2_k \to P^2_k$ 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 $P^2_k$ sends C to D. However it is possible with a birational isomorphism.