There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations the construction is short, elementary, and explicit.Our methods rely on simple algebraic constructions. It gives explicit expressions of members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. In particular, it explains the connections with commuting schemes of matrices. We also give some examples showing how our method can be used.There are many beautiful constructions of the Hilbert scheme of closed subschemes of projective schemes. In many cases where Hilbert schemes are involved, it suffices to know they exist. On the other hand, there are many situations when it is crucial to have an explicit description of the Hilbert schemes. In this article we give a simple construction of Hilbert schemes of points, under general circumstances, that provides such a description. In addition to being useful for computations, the construction is short, elementary and explicit.The main novelty of this article is the construction of the Hilbert scheme of n points of an affine scheme Spec(R) over an affine base scheme Spec(A), where R is any A-algebra. As a consequence of this construction we obtain a description, and easy proof of the existence, of the Hilbert scheme of n points of a family Proj(R) → S over an arbitrary base scheme S, where R is any quasi-coherent graded O S -algebra. We give explicit expressions for members of an affine covering of the Hilbert schemes, involving few variables satisfying natural equations. This provides a

Abstract. We give an explicit construction, for a flat map X → S of algebraic spaces, of an ideal in the n'th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemes of the locus of n distinct points. This generalises Haiman's corresponding result ([13]) for the affine complex plane. However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory.In the non-flat case we obtain a similar result by replacing the n'th symmetric product by the n'th divided power product.The Hilbert scheme, Hilb n X/S , of length n subschemes of a scheme X over some S is in general not smooth even if X → S itself is smooth. Even worse, it may not even be (relatively) irreducible. In the case of the affine plane over the complex numbers (where the Hilbert scheme is smooth and irreducible) Haiman (cf., [13]) realised the Hilbert scheme as the blow-up of a very specific ideal of the n'th symmetric product of the affine plane. It is the purpose of this article to generalise Haiman's construction. As the Hilbert scheme in general is not irreducible while the symmetric product is (for a smooth geometrically irreducible scheme over a field say) it does not seem reasonable to hope to obtain a Haiman like description of all of Hilb n X/S and indeed we will only get a description of the schematic closure of the open subscheme of n distinct points. With this modification we get a general result which seems very close to that of Haiman. The main difference from the arguments of Haiman is that we need to define the ideal that we want to blow up in a general situation and Haiman's construction seems to be too closely tied to the 2-dimensional affine space in characteristic zero.As a bonus we get that our constructions work very generally. We have thus tried to present our results in a generality that should cover reasonable applications (encouragement from the referee has made us make it more general than we did in a previous version of this article).There are some rather immediate consequences of this generality. The first one is that we have to work with algebraic spaces instead 2000 Mathematics Subject Classification. 14C05.

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.

Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.