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

TL;DR

This paper investigates two conjectures related to the dimension of the Fano scheme of lines on hypersurfaces, reducing the problem to a question about the intersection of top Chern classes of specific vector bundles.

Contribution

It reduces the Debarre-de Jong and Beheshti-Starr conjectures to a problem involving the intersection theory of vector bundles, providing a new approach to these longstanding conjectures.

Findings

The conjectures are equivalent to a non-vanishing intersection problem.

The reduction links geometric properties of hypersurfaces to algebraic intersection theory.

This approach offers a new pathway for proving or disproving the conjectures.

Abstract

We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the dimension of the Fano scheme of lines for hypersurfaces of degree at least n in n-dimensional projective space, reduce to determining if the intersection of the top Chern classes of certain vector bundles is nonzero.