Resultants and the Hilbert scheme of points on the line

Abstract

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.

Norms on rings and the Hilbert scheme of points on the line

“…Further explorations of the Spectral Mapping Theorem led to a short and simple proof of a generalized version of this result, valid for norms on algebras; see [12]. In the present article we generalize the algebraic results of [10], [17]. Our point of view as well as the algebraic techniques that we use are completely different from those of the articles [10], [17].…”

“…In the present article we generalize the algebraic results of [10], [17]. Our point of view as well as the algebraic techniques that we use are completely different from those of the articles [10], [17]. The methods are more general and allow us to obtain shorter and more transparent proofs of the results of these articles and to minimize the role of norms, and the use of the Generalized Spectral Mapping Theorem of [12].…”

“…In a previous article we proved the existence of an affine Hilbert scheme that parametrizes finite closed subschemes of length n of the affine scheme whose coordinate ring is the local ring of the affine line at the origin, and we gave an explicit description of the Hilbert scheme. The results on the representation and structure of the Hilbert scheme were generalized in [17] to the case of finite length subschemes of the affine scheme whose coordinate ring is any localization of the polynomial ring k½x in a variable x over any base ring k. Instead of generalizing the special algebraic tools used in [10], the article [17] used the Spectral Mapping Theorem. Further explorations of the Spectral Mapping Theorem led to a short and simple proof of a generalized version of this result, valid for norms on algebras; see [12].…”

“…One of the most interesting features of the results of [17,10] is that they show that one should take extreme care in studying only the rational points of Hilbert schemes. Indeed, the Hilbert schemes that are described in [17,10] are generally of high dimension but have extremely few, and sometimes no, rational points.…”

An elementary, explicit, proof of the existence of Hilbert schemes of points

“…To illustrate the flexibility of our methods we compute the Hilbert scheme of the scheme Spec(S −1 A[X ]) over Spec(A), where S is a multiplicatively closed subscheme of the polynomial ring A[X ] in the variable X over A (see [10,11,17]). …”