Automorphisms of generic hyperplane sections
$begingroup$
Let $Xsubset mathbb {P}^n=mathbb {CP}^n$ be a smooth hypersurface of degree $d$, ${H_lambda}_{lambdain {mathbb P^n}}$ be the set of hyperplane sections of $X$. We exclude the case $(d,n-2)=(4,2)$ or $(3,1)$ to ensure the automorphism of $H$ preserve polarizations, and also exclude the trivial case $nleq 2$. I want to know if the following is true:
For generic $lambda$, $Aut(H_lambda)=id$.
Easy to see it is enough to show there exist one $lambda$ with $Aut(H_lambda)=id$, but it is still unknown to me. Is this some known fact?
I am aware of the fact that generic hypersurfaces (or more general, generic complete intersections) have trivial automorphism group. But I didn't see how to relate it with this.
algebraic-geometry intersection-theory
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
$endgroup$
add a comment |
$begingroup$
Let $Xsubset mathbb {P}^n=mathbb {CP}^n$ be a smooth hypersurface of degree $d$, ${H_lambda}_{lambdain {mathbb P^n}}$ be the set of hyperplane sections of $X$. We exclude the case $(d,n-2)=(4,2)$ or $(3,1)$ to ensure the automorphism of $H$ preserve polarizations, and also exclude the trivial case $nleq 2$. I want to know if the following is true:
For generic $lambda$, $Aut(H_lambda)=id$.
Easy to see it is enough to show there exist one $lambda$ with $Aut(H_lambda)=id$, but it is still unknown to me. Is this some known fact?
I am aware of the fact that generic hypersurfaces (or more general, generic complete intersections) have trivial automorphism group. But I didn't see how to relate it with this.
algebraic-geometry intersection-theory
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
$endgroup$
$begingroup$
You should also exclude the case $d le 2$, when the automorphisms group is definitely nontrivial.
$endgroup$
– Sasha
Dec 27 '18 at 6:44
$begingroup$
@Sasha Yes, I forgot to exclude the trivial case. I added this in the text.
$endgroup$
– Akatsuki
Dec 28 '18 at 0:30
add a comment |
$begingroup$
Let $Xsubset mathbb {P}^n=mathbb {CP}^n$ be a smooth hypersurface of degree $d$, ${H_lambda}_{lambdain {mathbb P^n}}$ be the set of hyperplane sections of $X$. We exclude the case $(d,n-2)=(4,2)$ or $(3,1)$ to ensure the automorphism of $H$ preserve polarizations, and also exclude the trivial case $nleq 2$. I want to know if the following is true:
For generic $lambda$, $Aut(H_lambda)=id$.
Easy to see it is enough to show there exist one $lambda$ with $Aut(H_lambda)=id$, but it is still unknown to me. Is this some known fact?
I am aware of the fact that generic hypersurfaces (or more general, generic complete intersections) have trivial automorphism group. But I didn't see how to relate it with this.
algebraic-geometry intersection-theory
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
$endgroup$
Let $Xsubset mathbb {P}^n=mathbb {CP}^n$ be a smooth hypersurface of degree $d$, ${H_lambda}_{lambdain {mathbb P^n}}$ be the set of hyperplane sections of $X$. We exclude the case $(d,n-2)=(4,2)$ or $(3,1)$ to ensure the automorphism of $H$ preserve polarizations, and also exclude the trivial case $nleq 2$. I want to know if the following is true:
For generic $lambda$, $Aut(H_lambda)=id$.
Easy to see it is enough to show there exist one $lambda$ with $Aut(H_lambda)=id$, but it is still unknown to me. Is this some known fact?
I am aware of the fact that generic hypersurfaces (or more general, generic complete intersections) have trivial automorphism group. But I didn't see how to relate it with this.
algebraic-geometry intersection-theory
algebraic-geometry intersection-theory
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
edited Dec 28 '18 at 0:32
Akatsuki
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
asked Dec 26 '18 at 23:34
AkatsukiAkatsuki
1,0841725
1,0841725
$begingroup$
You should also exclude the case $d le 2$, when the automorphisms group is definitely nontrivial.
$endgroup$
– Sasha
Dec 27 '18 at 6:44
$begingroup$
@Sasha Yes, I forgot to exclude the trivial case. I added this in the text.
$endgroup$
– Akatsuki
Dec 28 '18 at 0:30
add a comment |
$begingroup$
You should also exclude the case $d le 2$, when the automorphisms group is definitely nontrivial.
$endgroup$
– Sasha
Dec 27 '18 at 6:44
$begingroup$
@Sasha Yes, I forgot to exclude the trivial case. I added this in the text.
$endgroup$
– Akatsuki
Dec 28 '18 at 0:30
$begingroup$
You should also exclude the case $d le 2$, when the automorphisms group is definitely nontrivial.
$endgroup$
– Sasha
Dec 27 '18 at 6:44
$begingroup$
You should also exclude the case $d le 2$, when the automorphisms group is definitely nontrivial.
$endgroup$
– Sasha
Dec 27 '18 at 6:44
$begingroup$
@Sasha Yes, I forgot to exclude the trivial case. I added this in the text.
$endgroup$
– Akatsuki
Dec 28 '18 at 0:30
$begingroup$
@Sasha Yes, I forgot to exclude the trivial case. I added this in the text.
$endgroup$
– Akatsuki
Dec 28 '18 at 0:30
add a comment |
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$begingroup$
You should also exclude the case $d le 2$, when the automorphisms group is definitely nontrivial.
$endgroup$
– Sasha
Dec 27 '18 at 6:44
$begingroup$
@Sasha Yes, I forgot to exclude the trivial case. I added this in the text.
$endgroup$
– Akatsuki
Dec 28 '18 at 0:30