Double-generic initial ideal and Hilbert scheme

Abstract

Following the approach in the book "Commutative Algebra", by D. Eisenbud, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the generic initial extensor of a subset of a Grassmannian and then the double-generic initial ideal of a so-called GL-stable subset of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric properties of a Hilbert scheme: they provide a necessary condition for a Borel ideal to correspond to a point of a given irreducible component, lower bounds for the number of irreducible components in a Hilbert scheme and the maximal Hilbert function in every irreducible component. Moreover, we prove that every isolated component having a smooth double-generic initial ideal is rational. As a byproduct, we prove that the Cohen-Macaulay locus of the Hilbert scheme parameterizing subschemes of codimension 2 is the union of open subsets isomorphic to affine spaces. This improves results by J.

Macaulay-like marked bases

“…If d > 1, then Mf(P(J)) can be embedded in a Hilbert scheme as an open subscheme by [9, Subsection 1.4 and Theorem 3.1] and we conclude. We can also observe that every quasi-stable saturated ideal J ⊂ A[x 0 , x] such that A[x 0 , x]/J is Cohen-Macaulay is the so-called double-generic initial ideal of the irreducible components containing J in a Hilbert scheme (see [8,Proposition 4(b) and Definition 5]).…”

Pathologies on the Hilbert scheme of points

“…Do all finite k-schemes, for a finite field k, lift to characteristic zero? Question 1.1 was completely open, and Hilb 4 (A 3 C ) is the only known reduced but singular Hilbert scheme of points [RA16,BCR17]. It was explicitly asked in [Ame10, Problem 1.6] whether Hilb 8 (A 4 C ) is reduced.…”

The Gröbner fan of the Hilbert scheme

“…The case of punctual Hilbert schemes has been studied continuously since the 70s (see [25] and references therein), and it is still under investigation nowadays [8,24,27,28,37]. In the case of 1-dimensional subschemes of the projective space P 3 there is a remarkable variety of results (for instance about ACM curves, see [13,42,14,5]).…”

“…In this context, a classical approach consists in trying to rephrase a global question in terms of a local question for a few, possibly finite, number of points of Hilb n p(t) . For instance, under the right conditions, the rationality of an irreducible component can be deduced by the smoothness of a special point lying on it [31,Corollary 6.10], [5,Theorem 6]. An efficient way to accomplish this task is to consider Gr öbner degenerations to monomial ideals and in particular to generic initial ideals.…”

“…Is it possible to deduce information about the topological structure of Hilb n p(t) ? These two problems are discussed in the inspiring paper "Double-generic initial ideal and Hilbert scheme" [5] by Bertone, Cioffi and Roggero. The double-generic initial ideal is a Borelfixed ideal associated to an irreducible component of Hilb n p(t) .…”