Cristina Bertone

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.

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.

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.

Functors of liftings of projective schemes

Abstract A classical approach to investigate a closed projective scheme W consists of considering a general hyperplane section of W, which inherits many properties of W. The inverse problem that consists in finding a scheme W starting from a possible hyperplane section Y is called a lifting problem, and every such scheme W is called a lifting of Y. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for Y is smooth also for W. We characterize all the liftings of Y with a given Hilbert polynomial by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. We use constructive methods from Grobner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.