Debarre–de Jong conjecture on lines on Fano hypersurfaces

Let $X\subset \mathbb{P}^n$ be a smooth hypersurface of degree $d$, and let $\mathcal{F}(X)$ denote its Fano variety of lines. A smooth hypersurface is Fano when $d\leq n$. The expected dimension of $\mathcal{F}(X)$ is $2n-d-3$. Debarre–de Jong conjecture. If $d\leq n$, then $\mathcal{F}(X)$ has the expected dimension.

The conjecture concerns the dimension of the variety of lines on smooth Fano hypersurfaces. It is known for $d\leq 8$ and, more generally, for $n\geq 2d-4$, while the full statement remains open.