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

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection ∇ :We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d ≤ 2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d ≥ 3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a Q-Gorenstein singularity of dimension d ≥ 2. We show that this condition is sufficient for the canonical module ω A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω A admits an integrable connection if and only if A is Gorenstein.Let k be an algebraically closed field of characteristic 0, and let A be a commutative k-algebra. For any A-module M, we consider connections on M, i.e. A-linear homomorphisms ∇ : Der k (A) → End k (M) such that ∇ D (am) = a∇ D (m) + D(a)m for all D ∈ Der k (A), a ∈ A and m ∈ M. A connection is integrable if it is a Lie algebra homomorphism. The present paper is devoted to the following question: Under what conditions on A and M is it possible to find a connection on M?

We consider the notion of a connection on a module over a commutative ring, and recall the obstruction calculus for such connections. The obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Grobner bases, we need the conversion to a description using free resolutions. We describe our implementation in Singular 3.0, available as the library conn.lib. Finally, we use the library to verify some known results and to obtain a new theorem for maximal Cohen-Macaulay (MCM) modules on isolated singularities. For a simple hypersurface singularity of dimension one or two, it is known that all MCM modules admit connections. We prove that for a simple threefold hypersurface singularity of type A_n, D_n or E_n, only the free MCM modules admit connections if n is at most 50.Comment: 14 pages, LaTeX, paper re-organized, new references added. To appear in Journal of Symbolic Computatio

Let k be an algebraically closed field of characteristic 0, let R be a commutative k-algebra, and let M be a torsion free R-module of rank one with a connection ∇. We consider the Lie-Rinehart cohomology with values in End R (M) with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on M. When R is an isolated singularity of dimension d 2, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when R is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.