Clustered families and applications to Lang‐type conjectures

Abstract

We introduce and classify 1-clustered families of linear spaces in the Grassmannian G ( k − 1 , n ) $\mathbb {G}(k-1,n)$ and give applications to Lang-type conjectures. Let X ⊂ P n $X \subset \mathbb {P}^n$ be a very general hypersurface of degree d $d$ . Let Z L $Z_L$ be the locus of points contained in a line of X $X$ . Let Z 2 $Z_2$ be the closure of the locus of points on X $X$ that are swept out by lines that meet X $X$ in at most 2 points. We prove that: if d ⩾ 3 n + 2 2 $d \geqslant \frac{3n+2}{2}$ , then X $X$ is algebraically hyperbolic modulo Z L ; $Z_L\hbox{\it ;}$ if d ⩾ 3 n 2 $d \geqslant \frac{3n}{2}$ , X $X$ contains lines but no other rational curves; if d ⩾ 3 n + 3 2 $d \geqslant \frac{3n+3}{2}$ , then the only points on X $X$ that are rationally Chow zero equivalent to points other than themselves are contained in Z 2 $Z_2$ ; if d ⩾ 3 n + 2 2 $d \geqslant \frac{3n+2}{2}$ and a relative Green–Griffiths–Lang Conjecture holds, then the exceptional locus for X $X$ is contained in Z 2 $Z_2$ . These sharpen prior results of Ein, Voisin, Pacienza, Clemens and Ran.