## Resultants and the Hilbert scheme of points on the line.(English)Zbl 1022.14003

The paper extends previous work by R. M. Skjelnes and 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 $$A[x]$$.

### MSC:

 14C05 Parametrization (Chow and Hilbert schemes) 14H05 Algebraic functions and function fields in algebraic geometry 14D22 Fine and coarse moduli spaces 13F20 Polynomial rings and ideals; rings of integer-valued polynomials

Zbl 1056.14500
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### References:

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