Double-Generic Initial Ideal and Hilbert Scheme

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).

On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that for all strongly stable ideals $J$, the marked schemes $mathbf{Mf}_{J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederers result about Grobner strata of zero-dimensional ideals, proving that Grobner strata of any ideals are locally closed subschemes of Hilbert schemes.

Flat families by strongly stable ideals and a generalization of Groebner bases

Let J be a strongly stable monomial ideal in S=K[x_1,...,x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs to Mf(J) if and only if it is generated by a special set of polynomials, the J-marked basis of I, that in some sense generalizes the notion of reduced Groebner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Groebner basis with respect to some term order, a sort of normal form modulo I (with the ideal I in Mf(J)) can be computed for every homogeneous polynomial, so that a J-marked basis can be characterized by a Buchberger-like criterion. Using J-marked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of affine scheme that turns out to be homogeneous with respect to a non-standard grading and flat in the origin (the point corresponding to J), thanks to properties of J-marked bases analogous to those of Groebner bases about syzygies.