Elementary components of Hilbert schemes of points

Consider the Hilbert scheme of points on a higher dimensional affine space. Its componentiselementaryif it parameterizes irreducible subschemes. We characterize reduced elementarycomponents in terms of tangent spaces and provide a computationally efficient way of findingsuch components. As an example, we find an infinite family of elementary and generically smoothcomponents on the affine four-space. We analyse singularities and formulate a conjecture whichwould imply the non-reducedness of the Hilbert scheme. Our main tool is a generalization of theBialynicki–Birula decomposition for this singular scheme.