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 $d\leq 2$. I want to know if the following is true:
For a general $\lambda\in \mathbb P^n$, ${\rm Aut}(H_\lambda)=id$.
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.