Infinite intersections of open subschemes and the Hilbert scheme of points

Abstract

We study infinite intersections of open subschemes and the corresponding infinite intersection of Hilbert schemes. If { U α } is the collection of open subschemes of a variety X containing the fixed point P, then we show that the Hilbert functor of flat and finite families of Spec( O X,P)=⋂ αU α is given by the infinite intersection ⋂ α Hilb U α , where Hilb U α is the Hilbert functor of flat and finite families on U α . In particular, we show that the Hilbert functor of flat and finite families on Spec( O X,P) is representable by a scheme.