Let $X\subset \mathbb P^n=\mathbb P^n_\mathbb C$ $(n>3)$ be a smooth hypersurface of degree $d\geq 3$, defined by a homogeneous polynomial $F$. Let $a=[a_0:\ldots:a_n]\in \mathbb P^n$ be any closed point, then the corresponding polar divisor with pole $a$ is defined to be the zero locus of $\sum a_i \partial_iF=0$ in $\mathbb P^n$. I believe the following statement is always true but I did not find how to prove:

there is no polar divisor contains a hyperplane in $\mathbb P^n$.

There is a very geometric interpretation: if a hyperplane contains in some polar divisor, then the nearby hyperplane sections of $X$ are all isomorphic. This phenomenon should not occur when degree is at least $3$. (In the case of degree $2$, a polar divisor is always a hyperplane, which should be interpreted as all smooth quadratic surfaces are isomorphic. This is also the classic pole and polar.)

Thanks in advance.


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