An elementary, explicit, proof of the existence of Hilbert schemes of points

Abstract

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

An elementary, explicit, proof of the existence of Quot schemes of points

“…We give here, for any morphism of schemes Proj() → S, where S is arbitrary and is a quasicoherent graded ᏻ Salgebra, an elementary construction of quotient schemes parametrizing equivalence classes of surjections from a quasicoherent ᏻ Proj() -module to coherent modules that are of relatively finite rank over S. The construction provides a natural and explicit description of an affine covering of the quotient schemes. In a previous article [Gustavsen et al 2007] we indicated the usefulness of such a description in the case of Hilbert schemes of points, and further evidence of this is given by M. Huibregtse [2002;. Our proof of the existence of the quotient schemes is a simplification and clarification of the constructions of these works.…”

“…The material of this section will basically give a construction of Hilbert schemes of points, as in [Gustavsen et al 2007], but the presentation here is different from that of that paper. We use that the A-algebra Sym A (G ⊗ A End A (E)ˇ) parametrizes module homomorphisms u : G → End A (E).…”

“…When Ᏺ = ᏻ X we obtain the existence and construction of the Hilbert scheme Hilb n X/S of n points in X , as in [Gustavsen et al 2007]. …”

Deformations of border bases

“…It is the only fiber in the family which is defined by a monomial ideal. Although it is known that the universal family is free with basis O (see [6,10]), we believe that the following proof which generalizes the method in [20] is very elementary and conceptually simple. Proof.…”

An elementary construction of the multigraded Hilbert scheme of points

“…In addition to the shift from description to construction, the present paper also goes beyond [Huibregtse 2002] by treating all cases n ≥ 1 in a uniform fashion, rather than just the case n = 2. [Gustavsen et al 2005] that presents another elementary proof of the existence of Hilbert schemes of points.…”