Giuseppe Valla

Hilbert functions of filtered modules

In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration.

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s ≤ 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

A sharp bound for the regularity index of fat points in general position

A bound is given for the regularity index of the coordinate ring of a set of fat points in general position in Pkn . The bound is attained by points on a rational normal curve.

Problems and Results on Hilbert Functions of Graded Algebras

Since a projective variety V = Z(I) ⊆ P n is an intersection of hypersurfaces, one of the most basic problems we can pose in relation to V is to describe the hypersurfaces containing it. In particular, one would like to know the maximal number of linearly independent hypersurfaces of each degree containing V, that is to know the dimension of I d , the vector space of homogeneous polynomials of degree d vanishing on V for various d. Since one knows the dimension \(\left( {\frac{{n + d}}{d}} \right)\) of the space of all forms of degree d, knowing the dimension of I d is equivalent to knowing the Hilbert function of the homogeneous coordinate ring A = k[X 0,..., X n ]/I of V, which is the vector space dimension of the degree d part of A.

Castelnuovo-Mumford regularity and extended degree

Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring A is effectively bounded by the dimension and any extended degree of A. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.

On set-theoretic complete intersections in the projective space

SuntoIn questo lavoro si considerano molte classi di varietà proiettive che sono insiemisticamente intersezione completa. Utilizzando la teoria delle basi di Gröbner, si deduce un metodo unificante di verifica della suddetta proprietà.SummaryIn this paper we consider many classes of projective varieties which are set-theoretically complete intersections; by using the theory of the Gröbner bases, we deduce an unifying method for the verification of such property.

Number of generators of ideals

Let I be a homogeneous ideal of a polynomial ring over a field, v(I ) the number of elements of any minimal basis of I, e = e(I ) the multiplicity or degree of R/I, h = h(I ) the height or codimension of I, i = indeg ( I ) the initial degree of J , i.e. the minimal degree of non zero elements of I . This paper is mainly devoted to find bounds for v(I ) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.

On the coefficients of the Hilbert polynomial

Let (A, m) be Cohen-Macaulay local ring with maximal ideal m and dimension d. It is well known that for n > 0, the length of the A-module A/mn is given by iAAmn=eontd−1d−e1n+d−2d−1+⋯+(−1)ded. The integers paper an ei are called the Hilbert coefficients of A. In this paper an upper bound is given for e2 in terms of e0, e1 and the embedded codimension h of A. If d ≤ 2 and the bound is reached, A has a specified Hilbert function. Similarly, in the one-dimensional case, we study the extremal behaviour with respect to the known inequality e1≤e2−h2.

On normal flatness and normal torsion-freeness☆

is primary for every 7t iff G(I) = @ P/P+l is a torsion-free R/I-module and therefore what we want to study is a sort of transitivity of normal torsion-freeness. This is strictly related to the transitivity of normal flatness and on this matter Hironaka (see [3, Theorem 31) proved an important theorem, which afterwards was improved by Grothendieck (see [l, IV, , 19.7.11). More recently Herrmann and Schmidt notably weakened the hypotheses but in their proof some rings still needed to be Cohen- Macaulay (see [2]). In the first section of this work we achieve two results: first we prove the theorem of transitivity of normal flatness in its widest generality (Theorem 1.5) and then an analogous theorem of transitivity of normal torsion-freeness (Theorem 1.6); although the analogy is not complete, suitable examples show that it cannot be strengthened. In the second section we deduce some consequences of more geometrical nature. The first one is an application to schemes of the theorem of transitivity of normal flatness; the second one is an answer to the problem we started from and allows us to get an interesting corollary in the case of projective schemes (Corollary 2.4). Finally, using this result, we can exhibit examples of projective curves such that the square of the corresponding prime ideals are not primary even if the curves are smooth and projectively Gorenstein. This gives an answer to a question raised by Hochster in [4].

The diagonal subalgebra of a blow-up algebra

Given a bigradedk-algebraS = ⊕(u,ν) S(u,ν), (u,ν) eℕ×ℕ, (k a field), one attaches to it the so-called diagonal subalgebraSδ = ⊕(u,u) S(u,u). This notion generalizes the concept of Segre product of graded algebras. The classical situation hasS = k[S(1,0), S(0,1)], whereby taking generators ofS(0,1) andS(0,1) yields a closed embedding Proj(S) ↪ℙkn − 1 ×ℙkr − 1, for suitablen,r; the resulting generators ofS(1,1) makeSδ isomorphic to the homogeneous coordinate ring of the image of Proj (S) under the Segre mapℙkn − 1 ×ℙkr − 1 →ℙknr − 1. The main results of this paper deal with the situation whereS is the Rees algebra of a homogeneous ideal generated by polynomials in a fixed degree. In this framework,Sδ is a standard graded algebra which, in some case, can be seen as the homogeneous coordinate ring of certain rational varieties embedded in projective space. This includes some examples of rational surfaces inpk5 and toric varieties inℙkn. The main concern is then with the normality and the Cohen-Macaulayness ofsδ. One can describe the integral closure ofsδ explicitly in terms of the given ideal and show that normality carries fromS toSδ. In contrast to normality, Cohen-Macaulayness fails to behave similarly, even in the case of the Segre product of Cohen-Macaulay graded algebras. The problem is rather puzzling, but one is able to treat a few interesting classes of ideals under which the corresponding Rees algebras yield Cohen-Macaulay diagonal subalgebras. These classes include complete intersections and determinantal ideals generated by the maximal minors of a generic matrix.