Double-generic initial ideal and Hilbert scheme

• Published in: Annali di matematica pura ed applicata 2017-02, Vol.196 (1), p.19-41
• Main Authors: Bertone, Cristina, Cioffi, Francesca, Roggero, Margherita
• Format: Article
• Language: English
• Subjects: Lower bounds; Mathematics; Mathematics and Statistics; Set theory
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-016-0560-0

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 by-product, 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 Fogarty (Am J Math 90:511–521, 1968 ) and Treger (J Algebra 125(1):58–65, 1989 ).