On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

Abstract

Conjecture 1.1. Let Xn−1 ⊂ P n be a hypersurface of degree d ≥ n and let F(X) ⊂ G(2, n+ 1) denote the Fano scheme of lines on X. Let B ⊂ F(X) be an irreducible component of dimension at least n − 2. Let IB := {(x,E) | x ∈ X, E ∈ B, x ∈ PE}, and let π and ρ denote (respectively) the projections to X and B. Let XB = π(IB) ⊆ X and let Cx = πρ−1ρπ−1(x). Then, for all x ∈XB , Cx ∩Xsing = ∅. If we take hyperplane sections in the case d = n, then Conjecture 1.1 would imply the following, which was conjectured independently by Debarre and de Jong.