  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.
