A Note on the Hilbert Scheme of Points on Lines in Higher Dimensions

This paper investigates the structure of the Hilbert scheme [Formula: see text], which parameterizes [Formula: see text]-point subschemes of a scheme [Formula: see text] formally equivalent to the union of the coordinate axes in [Formula: see text]. Using deformation theory, we provide explicit local defining equations for [Formula: see text] at each point, giving a constructive description of its scheme structure.