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 $m\geq m_0$

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

Let p(t) be an admissible Hilbert polynomial in P n of degree d. The Hilbert scheme Hilb n p(t) can be realized as a closed subscheme of a suitable Grassmannian G, hence it could be globally defined by homogeneous equations in the Plücker coordinates of G and covered by open subsets given by the non-vanishing of a Plücker coordinate, each embedded as a closed subscheme of the affine space A D , D = dim(G). However, the number E of Plücker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of Hilb n p(t) , we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than E. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree ≤ d + 2 in their natural embedding in A D . Furthermore we find new embeddings in affine spaces of far lower dimension than D, and characterize those that are still defined by equations of degree ≤ d + 2. The proofs are constructive and use a polynomial reduction process, similar to the one for Gröbner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.

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.

The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring k[x 0 , . . . , xn], in order to design two algorithms: the first one takes as input n and an admissible Hilbert polynomial P (z), and outputs the complete list of saturated quasi-stable ideals in the chosen polynomial ring with the given Hilbert polynomial. The second algorithm has an extra input, the characteristic of the field k, and outputs the complete list of saturated Borel-fixed ideals in k[x 0 , . . . , xn] with Hilbert polynomial P (z). The key tool for the proof of both algorithms is the combinatorial structure of a quasi-stable ideal, in particular we use a special set of generators for the considered ideals, the Pommaret basis.2010 Mathematics Subject Classification. 13P10, 14Q20, 12Y05, 05E40.