Clustered families and applications to Lang-type conjectures

TL;DR

This paper classifies 1-clustered families of linear spaces in Grassmannians and applies these results to establish new bounds and properties related to Lang-type conjectures for hypersurfaces in projective space.

Contribution

It introduces a classification of 1-clustered families and derives new geometric and hyperbolicity results for very general hypersurfaces based on degree bounds.

Findings

For degree d ≥ (3n+2)/2, hypersurfaces are algebraically hyperbolic outside lines.

For degree d ≥ (3n)/2, hypersurfaces contain lines but no other rational curves.

For degree d ≥ (3n+3)/2, points rationally equivalent to others are contained in loci swept out by specific lines.

Abstract

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 $Z_2$.