2020/10/21 by İzzet Coşkun, Coskun, Izzet, Eric Riedl +1 · 2 citations
Mathematics · #14G05 #14J70. Secondary: 14M15 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Primary: 32Q45
paper · pdf · doi:10.48550/arxiv.2010.11301
openalex publication_date 2020/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and classify 1-clustered families of linear spaces in the Grassmannian \mathbbG(k-1,n) and give applications to Lang-type conjectures. Let X ⊂ ℙn be a very general hypersurface of degree d. Let ZL be the locus of points contained in a line of X. Let Z2 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 ≥ (3n+2)/(2), then X is algebraically hyperbolic outside ZL. 2) If d ≥ (3n)/(2), X contains lines but no other rational curves 3) If d ≥ (3n+3)/(2), then the only points on X that are rationally Chow zero equivalent to points other than themselves are contained in Z2. 4) If d ≥ (3n+2)/(2) and a relative Green-Griffiths-Lang Conjecture holds, then the exceptional locus for X is contained in Z2.