On the families of hyperplane sections of some smooth projective varieties

In this note, we give two applications of \cite[Theorem 3.1]{Hwang}. We first study the free family $\mathcal{K}$ of hyperplane sections of the smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d\ge 3$. We prove that $X$ is determined by the free family $\mathcal{K}$ if $\dim(X)\ge 4$. As an application, we deduce that for $n\ge 4$, the hyperplane section of $X$ varies maximally in the moduli space of the smooth hypersurface of degree $d\ge 3$ in $\mathbb{P}^n$. We then study the free family of hyperplane sections of the smooth projective surface $X$ with Kodaira dimension $\kappa(X)\ge 0$. We prove that $X$ is determined by this free family.