Mixing
Automorphism of an open subset with completement of codimension $2$
$begingroup$
Let $mathbb P^n=mathbb {CP^n}$, I guess the following is true:
Let $Dsubset mathbb P^n$ be a closed subscheme of codimension $2$. Then every automorphism of $mathbb P^n-D$ is linear, i.e.
${rm Aut}(mathbb P^n-D)subset {rm Aut}(mathbb P^n)$.
More generally, I guess the following is also true:
Let $Dsubset V$ be a closed subscheme of codimension $2$, here $V$ is an arbitrary variety. Then ${rm Aut}(V-D)subset {rm Aut}(V)$.
Is it correct? Could someone give a reference or counter example about this?
I think the codimension $2$ condition is for Hartog's theorem, but I do not know how to apply it. Clearly we can not expect any morphism to extend to $D$, as there may be base locus, so we really need automorphisms. Also, it is not enough if $D$ is of codimension $1$, since there is trivial counter-example $mathbb P^n-D=mathbb A^n$.
algebraic-geometry reference-request complex-geometry automorphism-group
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
$endgroup$
add a comment |
$begingroup$
Let $mathbb P^n=mathbb {CP^n}$, I guess the following is true:
Let $Dsubset mathbb P^n$ be a closed subscheme of codimension $2$. Then every automorphism of $mathbb P^n-D$ is linear, i.e.
${rm Aut}(mathbb P^n-D)subset {rm Aut}(mathbb P^n)$.
More generally, I guess the following is also true:
Let $Dsubset V$ be a closed subscheme of codimension $2$, here $V$ is an arbitrary variety. Then ${rm Aut}(V-D)subset {rm Aut}(V)$.
Is it correct? Could someone give a reference or counter example about this?
I think the codimension $2$ condition is for Hartog's theorem, but I do not know how to apply it. Clearly we can not expect any morphism to extend to $D$, as there may be base locus, so we really need automorphisms. Also, it is not enough if $D$ is of codimension $1$, since there is trivial counter-example $mathbb P^n-D=mathbb A^n$.
algebraic-geometry reference-request complex-geometry automorphism-group
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
$endgroup$
$begingroup$
If $V$ is an affine normal variety and $Dsubset V$ is a closed subscheme of codimension at least two, then every morphism $Vsubset D to V$ extends (uniqutely) to a morphism $Vto V$ by Hartog's theorem. In particular, $mathrm{Aut}(V-D) = mathrm{Aut}(V)$ under these assumptions.
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:37
$begingroup$
Dear @AriyanJavanpeykar: (The automorphism should send $D$ to $D$, so we only get ${rm Aut}(V-D)subset {rm Aut}(V)$, but this is not important). Yes, but I am mainly interested in the case $V=mathbb P^n$, so we do not have the affineness.
$endgroup$
– Akatsuki
Jan 8 at 16:05
add a comment |
$begingroup$
Let $mathbb P^n=mathbb {CP^n}$, I guess the following is true:
Let $Dsubset mathbb P^n$ be a closed subscheme of codimension $2$. Then every automorphism of $mathbb P^n-D$ is linear, i.e.
${rm Aut}(mathbb P^n-D)subset {rm Aut}(mathbb P^n)$.
More generally, I guess the following is also true:
Let $Dsubset V$ be a closed subscheme of codimension $2$, here $V$ is an arbitrary variety. Then ${rm Aut}(V-D)subset {rm Aut}(V)$.
Is it correct? Could someone give a reference or counter example about this?
I think the codimension $2$ condition is for Hartog's theorem, but I do not know how to apply it. Clearly we can not expect any morphism to extend to $D$, as there may be base locus, so we really need automorphisms. Also, it is not enough if $D$ is of codimension $1$, since there is trivial counter-example $mathbb P^n-D=mathbb A^n$.
algebraic-geometry reference-request complex-geometry automorphism-group
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
$endgroup$
Let $mathbb P^n=mathbb {CP^n}$, I guess the following is true:
Let $Dsubset mathbb P^n$ be a closed subscheme of codimension $2$. Then every automorphism of $mathbb P^n-D$ is linear, i.e.
${rm Aut}(mathbb P^n-D)subset {rm Aut}(mathbb P^n)$.
More generally, I guess the following is also true:
Let $Dsubset V$ be a closed subscheme of codimension $2$, here $V$ is an arbitrary variety. Then ${rm Aut}(V-D)subset {rm Aut}(V)$.
Is it correct? Could someone give a reference or counter example about this?
I think the codimension $2$ condition is for Hartog's theorem, but I do not know how to apply it. Clearly we can not expect any morphism to extend to $D$, as there may be base locus, so we really need automorphisms. Also, it is not enough if $D$ is of codimension $1$, since there is trivial counter-example $mathbb P^n-D=mathbb A^n$.
algebraic-geometry reference-request complex-geometry automorphism-group
algebraic-geometry reference-request complex-geometry automorphism-group
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
asked Jan 7 at 16:02
AkatsukiAkatsuki
1,0791725
1,0791725
$begingroup$
If $V$ is an affine normal variety and $Dsubset V$ is a closed subscheme of codimension at least two, then every morphism $Vsubset D to V$ extends (uniqutely) to a morphism $Vto V$ by Hartog's theorem. In particular, $mathrm{Aut}(V-D) = mathrm{Aut}(V)$ under these assumptions.
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:37
$begingroup$
Dear @AriyanJavanpeykar: (The automorphism should send $D$ to $D$, so we only get ${rm Aut}(V-D)subset {rm Aut}(V)$, but this is not important). Yes, but I am mainly interested in the case $V=mathbb P^n$, so we do not have the affineness.
$endgroup$
– Akatsuki
Jan 8 at 16:05
add a comment |
$begingroup$
If $V$ is an affine normal variety and $Dsubset V$ is a closed subscheme of codimension at least two, then every morphism $Vsubset D to V$ extends (uniqutely) to a morphism $Vto V$ by Hartog's theorem. In particular, $mathrm{Aut}(V-D) = mathrm{Aut}(V)$ under these assumptions.
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:37
$begingroup$
Dear @AriyanJavanpeykar: (The automorphism should send $D$ to $D$, so we only get ${rm Aut}(V-D)subset {rm Aut}(V)$, but this is not important). Yes, but I am mainly interested in the case $V=mathbb P^n$, so we do not have the affineness.
$endgroup$
– Akatsuki
Jan 8 at 16:05
$begingroup$
If $V$ is an affine normal variety and $Dsubset V$ is a closed subscheme of codimension at least two, then every morphism $Vsubset D to V$ extends (uniqutely) to a morphism $Vto V$ by Hartog's theorem. In particular, $mathrm{Aut}(V-D) = mathrm{Aut}(V)$ under these assumptions.
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:37
$begingroup$
If $V$ is an affine normal variety and $Dsubset V$ is a closed subscheme of codimension at least two, then every morphism $Vsubset D to V$ extends (uniqutely) to a morphism $Vto V$ by Hartog's theorem. In particular, $mathrm{Aut}(V-D) = mathrm{Aut}(V)$ under these assumptions.
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:37
$begingroup$
Dear @AriyanJavanpeykar: (The automorphism should send $D$ to $D$, so we only get ${rm Aut}(V-D)subset {rm Aut}(V)$, but this is not important). Yes, but I am mainly interested in the case $V=mathbb P^n$, so we do not have the affineness.
$endgroup$
– Akatsuki
Jan 8 at 16:05
$begingroup$
Dear @AriyanJavanpeykar: (The automorphism should send $D$ to $D$, so we only get ${rm Aut}(V-D)subset {rm Aut}(V)$, but this is not important). Yes, but I am mainly interested in the case $V=mathbb P^n$, so we do not have the affineness.
$endgroup$
– Akatsuki
Jan 8 at 16:05
add a comment |
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$begingroup$
If $V$ is an affine normal variety and $Dsubset V$ is a closed subscheme of codimension at least two, then every morphism $Vsubset D to V$ extends (uniqutely) to a morphism $Vto V$ by Hartog's theorem. In particular, $mathrm{Aut}(V-D) = mathrm{Aut}(V)$ under these assumptions.
$endgroup$
– Ariyan Javanpeykar
Jan 8 at 13:37
$begingroup$
Dear @AriyanJavanpeykar: (The automorphism should send $D$ to $D$, so we only get ${rm Aut}(V-D)subset {rm Aut}(V)$, but this is not important). Yes, but I am mainly interested in the case $V=mathbb P^n$, so we do not have the affineness.
$endgroup$
– Akatsuki
Jan 8 at 16:05