Deformations of the Fano scheme of a cubic

We study the deformation theory of the Fano scheme $\mathrm{F}=\mathrm{F}(\mathrm{X})$ of lines on a cubic $\mathrm{X}$ of dimension $d$ with only finitely many singularities. By taking the relative Fano scheme, we define a morphism $\eta:\mathscr{D}_{\mathrm{X}}\rightarrow\mathscr{D}_{\mathrm{F}}$ of the local moduli functors associated to $\mathrm{X}$ and $\mathrm{F}$, respectively. We show that for $d\geqslant 5$, $\eta$ yields an isomorphism on first-order deformations; in particular, $\eta$ is an isomorphism whenever $\mathrm{H}^{0}(\Theta_{\mathrm{X}})=0$.