Roy Mikael Skjelnes

Resultants and the Hilbert scheme of points on the line

We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction ringsk[X]U of the one-variable polynomial ring over a commutative ringk. Our description is based on the computation of the resultant of polynomials ink[X]. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.

Notes on flatness and the Quot functor on rings

For flat modules M over a ring A we study the similarities between the three statements,dim k (P) ( k (P)⊗ A M =dfor all prime ideals P of A, the Ap-module M p is free of rank d for all prime ideals P of A, and M is a locally free J4-module of rank d. We have particularly emphasized the case when there is an>l-algebra B, essentially of finite type, and M is a finitely generated B-module.

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 {Uα} 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 )=⋂ α U α is given by the infinite intersection ⋂ α H ilb U α , where H ilb U α is the Hilbert functor of flat and finite families on Uα. In particular, we show that the Hilbert functor of flat and finite families on Spec ( O X,P ) is representable by a scheme.