Deformations of the Fano scheme of a cubic

TL;DR

This paper investigates the deformation theory of the Fano scheme of lines on a cubic hypersurface, establishing conditions under which the deformation functor of the Fano scheme is isomorphic to that of the cubic.

Contribution

It introduces a morphism between local moduli functors of the cubic and its Fano scheme and proves it induces an isomorphism on first-order deformations for dimensions at least 5.

Findings

For d ≥ 5, the morphism η induces an isomorphism on first-order deformations.

When H^0(Θ_X)=0, η is an isomorphism.

The study links the deformation theories of the cubic and its Fano scheme.

Abstract

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 $\mathrm{H}^{0}(\Theta_{\mathrm{X}})=0$.