Francesca Cioffi

BIVARIATE HERMITE INTERPOLATION AND LINEAR SYSTEMS OF PLANE CURVES WITH BASE FAT POINTS

. When the multiplicities are all equal,to m say, this problem has been attacked by a number of authors (Lorentz andLorentz, Ciliberto and Miranda, Hirschowitz) and there are a number of goodconjectures (Hirschowitz, Ciliberto and Miranda) on the dimension of these inter-polating spaces. The determination of the dimension has been already solved form ≤ 12 and all d and n by a degeneration technique and some ad hoc geometricarguments. Here this technique is applied up through m = 20; since it fails insome cases, we resort (in these exceptional cases) to the bivariete Hermite interpo-lation with the support of a simple idea suggested by Gr¨obner bases computation.In summary we are able to prove that the dimension of the vector space is theexpected one for 13 ≤ m ≤ 20.

Minimally Generating Ideals of Rational Parametric Curves in Polynomial Time

We present an algorithm for computing a minimal set of generators for the ideal of a rational parametric projective curve in polynomial time. The method exploits the availability of polynomial algorithms for the computation of minimal generators of an ideal of points and is an alternative to the existing Grobner bases techniques for the implicitization of curves. The termination criterion is based on the Castelnuovo?Mumford regularity of a curve. The described computation also yields the Hilbert function and, hence, the Hilbert polynomial and the Poincare series of the curves. Moreover, it can be applied to unions of rational curves. We have compared the implementation of our algorithm with the Hilbert driven elimination algorithm included in CoCoA 3.6 and Singular 1.2, obtaining, in general, significant improvements in timings.

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.

Regularity bounds by minimal generators and Hilbert function

LetρC be the regularity of the Hilbert function of a projective curveC inPKn over an algebraically closed fieldK andβ1,...,βn-1 be degrees for which there exists a complete intersection of type (β1,...,βn-1) containing properlyC. Then the Castelnuovo-Mumford regularity ofC is bounded above by max {ρC + 1,β1 +...+βn-1-(n-1)}. We investigate the sharpness of the above bound, which is achieved by curves algebraically linked to ones having degenerate general hyperplane section.

Computation of minimal generators of ideals of fat points

We present an implementation of an algorithm for computing minimal homogeneous generators and Hilbert functions of ideals of fat points. First we construct a set of generators by an interpolating method which is based on the Hermite interpolation and generalizes the algorithm of [20] to projective space. Then, we minimalize the generators by the same algorithm which has been developed in [6] by exploiting results of [23] and [19] and which has been already successfully applied to ideals of points, to ideals of rational curves [3] and to ideals of general parametric varieties [22].

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.

Geometric Description of Lifting Monomial Ideals

Abstract By focusing our attention on the set of monomials outside a given monomial ideal, we tackle the study of the geometric configurations of (reduced) unions of projective linear varieties arising from lifting monomial ideals via a classic lifting procedure, called t-lifting, and a more general lifting procedure, called pseudo-t-lifting. We observe that, in contrast to the Artinian case, in the positive dimensional case we may not obtain generalized stick figures also via a generic pseudo-t-lifting. In particular, in dimension 1 a generic pseudo-1-lifting produces a seminormal union of lines. Then we give conditions to obtain generalized stick figures by means of pseudo-t-liftings of non-Artinian monomial ideals. #Communicated by C. Pedrin.

Computing Minimal Generators of Ideals of Elliptic Curves

In this paper results about the Hilbert function and about the number of minimal gener- ators stated in (Orecchia) for disjoint unions of rational smooth curves are generalized to disjoint unions of distinct smooth non special curves. Hence, the maximal rank and the minimal generation of such curves are studied. In particular, we consider elliptic curves and we describe a method to compute their Hilbert functions in any dimension and for every choice of the degrees. Applications to the study of elliptic curves on threefolds are shown.

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.