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. Fogarty (1968) and R. Treger (1989).
Cite
@article{arxiv.1503.03768,
title = {Double-Generic Initial Ideal and Hilbert Scheme},
author = {Cristina Bertone and Francesca Cioffi and Margherita Roggero},
journal= {arXiv preprint arXiv:1503.03768},
year = {2016}
}
Comments
23 pages. Final version. The present version contains several changes, more precisely: correcting typos; a more complete Introduction, including new references to literature; improving the statements of Theorems 3.3 and 5.4 and adding details in Example 2.8 (numbering of this version)