GIT stable cubic threefolds and certain fourfolds of $K{3}^{[2]}$-type

Abstract

We study the behaviour on some nodal hyperplanes of the isomorphism, described by Boissière–Camere–Sarti, between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of $K{3}^{[2]}$-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6; along those hyperplanes, the automorphism degenerates by jumping to another family. We generalize their result to singular nodal cubic threefolds having one singularity of type $A_i$, for $i=2,3,4$, providing birational maps between the loci of cubic threefolds where a generic element has an isolated singularity of the types $A_i$ and some moduli spaces of hyperkähler fourfolds of $K{3}^{[2]}$-type with non-symplectic automorphism of order three belonging to different families. In order to treat the $A_2$ case, we introduce the notion of Kähler cone sections of $K$-type, generalizing the definition of $K$-general polarized hyperkähler manifolds.