An elementary construction of the multigraded Hilbert scheme of points

Abstract

We present an elementary construction of the multigraded Hilbert scheme of d points of A n k = Spec(k[x 1 ,..., X n ]), 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 × k A n k 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.