We present an elementary
construction of the multigraded Hilbert scheme of d points of 𝔸kn=Spec(k[x1,…,xn]),
where k is an arbitrary commutative and unitary ring. This Hilbert scheme represents
the functor from k-schemes to sets that associates to each k-scheme T the set of
closed subschemes Z ⊆ T ×k𝔸kn such that the direct image (via the first
projection) of the structure sheaf of Z is locally free of rank d on T. It is a special
case of the general multigraded Hilbert scheme constructed by Haiman and
Sturmfels. Our construction proceeds by gluing together affine subschemes
representing subfunctors that assign to T the subset of Z such that the
direct image of the structure sheaf on T is free with a particular set of d
monomials as basis. The coordinate rings of the subschemes representing the
subfunctors are concretely described, yielding explicit local charts on the Hilbert
scheme.
