On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces   with too many lines
  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. 
J.M. Landsberg and Orsola Tommasi