Deformations of the connected sum of Gorenstein algebras

We prove that the Gorenstein locus of the Hilbert scheme of points on ${\mathbb{A}}^n$ is non-reduced for $n\ge 12$; we construct examples of non-reduced points that come from apolar algebras of the sum of general cubics. As a corollary, we get a non-reducedness result for the cactus scheme. We generalise the Białynicki-Birula decomposition to abstract deformation functors, providing a new method of studying deformation theory. Our construction gives us fractal structures on the nested Hilbert scheme.