Clustered families and applications to Lang-type conjectures

We introduce and classify 1-clustered families of linear spaces in the Grassmannian $\mathbb{G}(k-1,n)$ and give applications to Lang-type conjectures. Let $X \subset \mathbb{P}^n$ be a very general hypersurface of degree $d$. Let $Z_L$ be the locus of points contained in a line of $X$. Let $Z_2$ be the locus of points on $X$ that are swept out by lines that meet $X$ in at most $2$ points. We prove that 1) If $d \geq \frac{3n+2}{2}$, then $X$ is algebraically hyperbolic outside $Z_L$. 2) If $d \geq \frac{3n}{2}$, $X$ contains lines but no other rational curves 3) If $d \geq \frac{3n+3}{2}$, then the only points on $X$ that are rationally Chow zero equivalent to points other than themselves are contained in $Z_2$. 4) If $d \geq \frac{3n+2}{2}$ and a relative Green-Griffiths-Lang Conjecture holds, then the exceptional locus for $X$ is contained in $Z_2$.