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Let $X\subset \mathbb {P}^n=\mathbb {CP}^n$ be a smooth hypersurface of degree $d$, $\{H_\lambda\}_{\lambda\in {\mathbb P^n}}$ 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 $n\leq 2$. 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.

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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

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