The Hilbert scheme of points and its link with border basis

Abstract

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 µ (P n ) are deduced from the commutation relations characterising border bases and from generating equations. We also prove that the punctual Hilbert functor Hilb µ P n can be represented by the scheme Hilb µ (P n ) 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.

Waring, tangential and cactus decompositions

“…In the setting of our algorithms the considered ideal I will be I Λ . By definition Λ ∈ I ⊥ Λ , hence by Theorem 2.7 we have decompositions of Λ as in (3). The proof of Theorem 2.7 given in [34] also shows that…”

“…The above quantity is usually called the multiplicity of ζ i . A decomposition as in (3) for which the sum of the multiplicities r = [22,13]) and such an r is referred to as the size of the generalized decomposition.…”

“…For this reason, in the unlucky case that a random choice of the h's does not lead to a required decomposition, then it is not sufficient to check other randomly chosen values of h's but a different component of V(I C2 ) has to be considered. After all these components had been tested we can conclude that r has to be increased in order to find a required decomposition since in [3,Theorem 3.16] it is proved that C 2 are quadratic equations for the punctual Hilbert scheme of points in P n of length r. This procedure works in general since the smooth elements in the Hilbert scheme of points of length r are generic in their connected component. Therefore this procedure provides an effective way to find a good choice of the h's and to decide that this cannot be achieved with the considered rank.…”

A general framework for Noetherian well ordered polynomial reductions

“…It follows then that the divisors of an element in the border are all contained in N(J) ∪ B(J) In many constructions of marked bases over the border, one considers a fixed term order and supposes that in each marked polynomial the elements in the tails are smaller than the head w.r.t. such a term order; anyway there also exist some bases, marked on B(J) without this constraint (see [59] and [1]).…”