Local Equations for Hilbert Schemes of Points

We compute the completion of the local ring of the Hilbert scheme of degree $n+1$ subschemes of ${\mathbb{A}}^n$ at the point corresponding to the ideal $⟨x_1,\dots ,x_n{⟩}^2$, and describe the completion of the universal family. For the purposes of comparison, we do this computation with both classical and DGLA methods. We use our explicit equations to produce high dimensional linear subspaces of the Hilbert scheme, and compare our equations with those coming from deformations of based algebras.