Clustered families and applications to Lang‐type conjectures

• Published in: Proceedings of the London Mathematical Society 2022-12, Vol.125 (6), p.1353-1376
• Main Authors: Coskun, Izzet, Riedl, Eric
• Format: Article
• Language: English
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms.12484

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⊂Pn$X \subset \mathbb {P}^n$ be a very general hypersurface of degree d$d$. Let ZL$Z_L$ be the locus of points contained in a line of X$X$. Let Z2$Z_2$ 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⩾3n+22$d \geqslant \frac{3n+2}{2}$, then X$X$ is algebraically hyperbolic modulo ZL;$Z_L\hbox{\it ;}$ if d⩾3n2$d \geqslant \frac{3n}{2}$, X$X$ contains lines but no other rational curves; if d⩾3n+32$d \geqslant \frac{3n+3}{2}$, then the only points on X$X$ that are rationally Chow zero equivalent to points other than themselves are contained in Z2$Z_2$; if d⩾3n+22$d \geqslant \frac{3n+2}{2}$ and a relative Green–Griffiths–Lang Conjecture holds, then the exceptional locus for X$X$ is contained in Z2$Z_2$. These sharpen prior results of Ein, Voisin, Pacienza, Clemens and Ran.