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

Abstract

Let J be a strongly stable monomial ideal in S = K[x 0 , . . . , xn] 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 Gröbner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Gröbner basis with respect to some term order, a sort of reduced form modulo I ∈ 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 Gröbner bases about syzygies.Definition 1.3.[34] A marked polynomial is a polynomial f ∈ S together with a specified term of Supp(f ) that will be called head term of f and denoted by Ht(f ). Definition 1.4. A finite set G of homogeneous marked polynomials f α = x α − c αγ x γ , with Ht(f α ) = x α , is called J-marked set if the head terms Ht(f α ) are pairwise different and form the monomial basis B J of a monomial ideal J and every x γ belongs to N (J), so that |Supp(f ) ∩ J| = 1. A J-marked set G is a J-marked basis if N (J) is a basis of S/(G) as a K-vector space, i.e. S = (G) ⊕ N (J) as a K-vector space.Remark 1.5. The ideal (G) generated by a J-marked basis G has the same Hilbert function as J, hence dim K J m = dim K (G) m for every m ≥ 0, by the definition of J-marked basis.Definition 1.6. The family of all homogeneous ideals I such that N (J) is a basis of the quotient S/I as a K-vector space will be denoted by Mf(J) and called J-marked family.Remark 1.7. (1) If I belongs to Mf(J), then I contains a J-marked set.(2) A J-marked family Mf(J) contains every homogeneous ideal having J as initial ideal with respect to some term order, but it can also contain other ideals, as we will see in Example 3.18.

Double-generic initial ideal and Hilbert scheme

“…≺ ≺ of the Borel ideals. By a direct computation involving marked schemes (see [16,7,9]), using for instance either the Singular library [13,17] or the algorithm described in [8], we obtain that b 4 is contained in two irreducible components Y 2 and Y 3 . Therefore, the point corresponding to b 4 is not smooth for the Hilbert scheme.…”

Computing Quot schemes via marked bases over quasi-stable modules

“…In this section, we extend the notions of marked polynomial, marked basis and marked family, investigated in [4,9,10,22] for ideals, to finitely generated modules in…”

“…The present paper is concerned with generalizing and deepening the results of [1,4,10,22] in order to investigate Quot schemes over fields of arbitrary characteristic. The Quot functor was introduced by Grothendieck in [16], where he also proved that this functor is the functor of points of a projective scheme.…”

A general framework for Noetherian well ordered polynomial reductions

“…Except [13,12,28] and Gordan 1 , all these results make a strong and non-necessary requirement in order to grant termination of the reduction procedures; in fact they imposed a semigroup ordering on the set of the monomials, i.e. an ordering that preserves multiplication by variables, while a noetherian well-founded ordering can be sufficient.…”

“…However such families of deformations in general cover only locally closed subschemes of Hilbert scheme and are not sufficient to study neighbourhoods of deformations of X , id est opens of Hilbert scheme; such opens can be obtained instead by considering [14] those ideals I of P which share with I a fixed monomial basis of the quotient P/I. In order to determine the family of all such ideals I of P, term-ordering free bases of polynomial ideals were introduced, under the label of marked bases in [13,12,28].…”