Margherita Roggero

Macaulay-like marked bases

We define marked sets and bases over a quasi-stable ideal 𝔧 in a polynomial ring on a Noetherian K-algebra, with K a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of 𝔧 and a given integer m. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough m, is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.

Term-ordering free involutive bases

In this paper, we consider a monomial ideal J ? P : = A x 1 , ? , x n , over a commutative ring A, and we face the problem of the characterization for the family M f ( J ) of all homogeneous ideals I ? P such that the A-module P / I is free with basis given by the set of terms in the Grobner escalier N ( J ) of J. This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes.For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced by Riquier (1893, 1899, 1910) and in Janet (1920, 1924, 1927) and we generalize the construction of J-marked bases and term-ordering free reduction process introduced and deeply studied in Bertone et al. (2013a), Cioffi and Roggero (2011) for the special case of a strongly stable monomial ideal J.Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F ( J ) , called stably complete, that allows an explicit description of the family M f ( J ) . We obtain stronger results if J is quasi-stable, proving that F ( J ) is a Pommaret basis and M f ( J ) has a natural structure of affine scheme.The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.

The Locus of Points of the Hilbert Scheme with Bounded Regularity

In this paper we consider the Hilbert scheme parameterizing subschemes of ℙ n with Hilbert polynomial p(t), and we investigate its locus containing points corresponding to schemes with regularity lower than or equal to a fixed integer r′. This locus is an open subscheme of and, for every s ≥ r′, we describe it as a locally closed subscheme of the Grasmannian given by a set of equations of degree ≤deg(p(t)) +2 and linear inequalities in the coordinates of the Plücker embedding.

Sui sistemi lineari e il gruppo delle classi di divisori di una varietà reale

SummaryIn this work we prove that the ringR[V] of an affine or projective real variety V is factorial if and only if the codimension 1 irreductible subvarieties of V are complete intersections. In particular ifR[V] is integrally closed, the divisor class group Cl (V) of the variety V is generated by the classes of the (homogeneous) real prime ideals ofR[V].

Minimal Castelnuovo–Mumford Regularity for a Given Hilbert Polynomial

Let K be an algebraically closed field of null characteristic and p(z) a Hilbert polynomial. We look for the minimal Castelnuovo–Mumford regularity mp(z) of closed subschemes of projective spaces over K with Hilbert polynomial p(z). Experimental evidences led us to consider the idea that mp(z) could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo–Mumford regularity mϱp(z) of schemes with Hilbert polynomial p(z) and given regularity ϱ of the Hilbert function, and also the minimal Castelnuovo–Mumford regularity mu of schemes with Hilbert function u. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.

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 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).

The scheme of liftings and applications

Abstract We study the locus of the liftings of a homogeneous ideal H in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme L H by applying the constructive methods of Grobner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of H. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed L H in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to finding interesting topological properties, as for instance that L H is connected and that its locus of radical liftings is open. Moreover, we show that every ideal defining an arithmetically Cohen–Macaulay scheme of codimension two has a radical lifting, giving in particular an answer to an open question posed by L.G. Roberts in 1989.

On curve sections of rank two reflexive sheaves

Let F be a normalized rank 2 reflexive sheaf on P3 with Chern classes c 1,c 2,c 3. Let α be the least integer such that 0≠H 0 F(α) and β be the smallest integer such that H 0 F(n) has sections whose zero scheme is a curve for all n≥ β . We show that if T 0 is the largest root of the cubic polynomial then β ≥ T 0-α-c 1-1. There are applications to the smallest degree of a surface containing a curves which are the zero schemes of sections of H 0 F(α).