Norms on rings and the Hilbert scheme of points on the line

We generalize the algebraic results of Laksov and Skjelnes (2001) and Skjelnes (2002), and obtain easy and transparent proofs of the representability of the Hilbert functor of points on the affine scheme whose coordinate ring is any localization of the polynomial ring in one variable over an arbitrary base ring. The coordinate ring of the Hilbert scheme is determined. We also make explicit the relation between our methods and the beautiful treatment of the Hilbert scheme of curves via norms, indicated by Grothendieck (1995), and performed by Deligne (1973).