Resultants and the Hilbert scheme of points on the line

Abstract

We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings k[X]_U of the one-variable polynomial ring over a commutative ring k. Our description is based on the computation of the resultant of polynomials in k[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.