The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F (x) of degree n in A[x] such that A ⊗k k[x](x)/(F (x)) is a free A–module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x). Introduction. We shall study the Hilbert scheme parameterizing finite length subschemes of the local ring k[x](x) of the line at the origin. The Hilbert schemes parameterizing finite length subschemes of local rings have mostly been studied for local rings at smooth points on surfaces (see e.g. [B], [BI], [C], [G], [I1], [I2], [I3], [P]). The focus has been on the rational points of the Hilbert schemes rather than on the schemes themselves. The purpose of the following work is to point out that we loose essential information about the Hilbert schemes parameterizing finite length subschemes of a local ring by considering rational points instead of families. Indeed, there is only one rational point k[x]/(x) of the Hilbert scheme parameterizing subschemes of k[x](x) of length n whereas, as we shall show in this article, the Hilbert scheme is affine of dimension n. The coordinate ring is equal to the localization of the symmetric polynomials of the ring k[t1, . . . , tn] in n variables, in the multiplicatively closed subset consisting of the products g(t1) · · · g(tn) for all polynomials g(x) in one variable over k such that g(0) 6= 0. In forthcoming work [S1] the second author will use the techniques and results of the present article to show that the functor of families with support at the origin, in contrast to the Hilbert functor, is not even representable. The functor of families with support at the origin is frequently used by some authors because it has the same rational points as the Hilbert scheme. In [S2] the second author shows how the techniques of the present article can be used on any localization of k[x], over any ring k, and gives the relation to the well known result that the Hilbert scheme of the projective line is given by the symmetric product. An easy and fundamental result in commutative algebra states that the residue ring A[x]/(F (x)) of the polynomial ring in a variable x over a ring A by the ideal generated by a monic polynomial F (x) of degree n, is a free A–module of rank n.