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 , where the Hilbert scheme on spectrum of the local ring of a point was described.
"Resultants and the Hilbert scheme of points on the line." Ark. Mat. 40 (1) 189 - 200, April 2002. https://doi.org/10.1007/BF02384509