# Automorphisms of generic hyperplane sections

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.

• You should also exclude the case $d \le 2$, when the automorphisms group is definitely nontrivial. – Sasha Dec 27 '18 at 6:44
• @Sasha Yes, I forgot to exclude the trivial case. I added this in the text. – Akatsuki Dec 28 '18 at 0:30