On a Conjecture on the Variety of Lines on a Fano Complete Intersection

The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in ${\mathbb{P}}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove that for a Fano complete intersection $X\subset {\mathbb{P}}^n$ of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on $X$ has the expected dimension.