Resultants and the Hilbert scheme of points on the line (Q1811521)

The paper extends previous work by \textit{R. M. Skjelnes} and \textit{D. Laksov} [Compos. Math. 126, 323-334 (2001; Zbl 1056.14500)] and gives a description of the Hilbert scheme of \(n\) points on the affine scheme \(C:=\text{Spec}(K[x]_U)\), where \(K\) is a field and \(K[x]_U\) is a fraction ring of the polynomial ring in one variable. The Hilbert functor of \(n\) points on \(C\), \(\text{Hilb}^n\), associates to a \(K\)-algebra \(A\) the set \(\text{Hilb}^n(A)\), formed by the ideals \(I\) of \(A\otimes{_KK}[x]_U\) such that the residue class ring \(A\otimes{_KK}[x]_U/I\) is locally free of finite rank \(n\) (as \(A\)-module). The main result is that \(\text{Hilb}^n\) is represented by the fraction ring \(H=K[s_1,...,s_n]_{U(n)}\), where \(s_1,...,s_n\) are the elementary symmetric functions in the variables \(t_1,...,t_n\) and \(U(n)=\{f(t_1)\dots f(t_n)\mid f\in U\}\). Moreover, the universal family of \(n\)-points on \(C\) is isomorphic to \(\text{Sym}_K^{n-1}(C)\times_K C\), as in the case where \(C\) is a smooth curve. The result relies on a characterization of the characteristic polynomial of the multiplication by the residue of a polynomial in \(A[x]/(F)\), where \(A\) is any commutative ring and \(F\) a monic polynomial in \(A[x]\).