Singularities of Fano varieties of lines on singular cubic fourfolds

TL;DR

This paper characterizes the singularities of Fano varieties of lines on singular cubic fourfolds, showing they mirror Hilbert schemes of points on ADE-singular surfaces, and identifies their symplectic resolutions.

Contribution

It establishes a link between singularities of Fano varieties on cubic fourfolds and Hilbert schemes on ADE surfaces, providing new insights into their structure and resolutions.

Findings

Fano varieties of lines on certain cubic fourfolds have ADE-type singularities.

These singularities are equivalent to those of Hilbert schemes of points on ADE surfaces.

The singularities admit unique symplectic resolutions.

Abstract

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities. This is shown as a corollary to the characterization of a singularity that is obtained as a $K3^{[2]}$-type contraction and has a unique symplectic resolution.