Singular Veronese double cones

Varieties with the largest possible number of isolated singularities in a given deformation family often have nice geometric properties. For instance, the Segre cubic, which is a cubic threefold with 10 nodes, is unique known to be related to certain moduli spaces of abelian surfaces. Del Pezzo threefolds with the maximal number of isolated singularities are double solids branched in Kummer quartic surfaces.
In my talk I will describe the geometry of del Pezzo threefolds of degree 1 (also known as Veronese double cones) with the maximal possible number of isolated singularities. Such varieties are nodal and have 28 singular points. They are in one-to-one correspondence with smooth plane quartics, and much of their properties can be recovered from the properties of these quartics.
The talk is based on a joint work in progress with H. Ahmadinezhad, I. Cheltsov and J. Park.