Let $J\subset S=K[x_0,\ldots,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme $\hilbp$, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the \emph{$m$-truncation} ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in $\Mf(\underline{J}_{\geq m})$ in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of $\Mf(\underline{J}_{\geq m})$ in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every $m$, we give a closed embedding $\phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1})$, characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$
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.
Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order in the Hilbert scheme of points in Pn. In this context, we look into the properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also on the connections between the shape of generators of Borel ideals and the related Hilbert polynomial, thus providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial
We define marked sets and bases over a quasi-stable ideal j in a polynomial ring on a\ud Noetherian K-algebra, with K a field of any characteristic. The involved polynomials\ud may be non-homogeneous, but their degree is bounded from above by the maximum\ud among the degrees of the terms in the Pommaret basis of j and a given integer m.\ud Due to the combinatorial properties of quasi-stable ideals, these bases behave well with\ud respect to homogenization, similarly to Macaulay bases. We prove that the family of\ud marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and,\ud for large enough m, is an open subset of a Hilbert scheme. Our main results lead to\ud algorithms that explicitly construct such a family. We compare our method with similar\ud ones and give some complexity results
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.
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