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

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