The Hilbert scheme of points and its link with border basis

This paper examines the effective representation of the punctual Hilbert scheme. We give new equations, which are simpler than Bayer and Iarrobino-Kanev equations. These new Plücker-like equations define the Hilbert scheme as a subscheme of a single Grassmannian and are of degree two in the Plücker coordinates. This explicit complete set of defining equations for Hilb(P) are deduced from the commutation relations characterising border bases and from generating equations. We also prove that the punctual Hilbert functor Hilb P can be represented by the scheme Hilb(P) defined by these relations and the well-known Plücker relations on the Grassmanian. A new description of the tangent space at a point of the Hilbert scheme, seen as a subvariety of the Grassmannian, is also given in terms of projections with respect to the underlying border basis.