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Let $X$ be a cubic fourfold that has only simple singularities and does notcontain a plane. We prove that the Fano variety of lines on $X$ has the sameanalytic type of singularity as the Hilbert scheme of two points on a surfacewith only ADE-singularities. This is shown as a corollary to thecharacterization of a singularity that is obtained as a $K3^{[2]}$-typecontraction and has a unique symplectic resolution.

Singularities of Fano varieties of lines on singular cubic fourfolds