CONFERENCE ON ZERO-DIMENSIONAL SCHEMES
AND RELATED TOPICS

June 6-8, 2002    Acireale, Italy (Sicily)
hotel / conference center La Perla Ionica

in honor of Tony Geramita on the occasion of his 60th birthday


Invited Speakers:


Abstracts

Maria Alberich-Carraminana

Enriques diagrams and adjacency of planar curve singularities

Enriques diagrams represent the equisingularity (or topological equivalence) classes of planar curve singularities. Given two equisingularity types, we study whether one of them is adjacent to the other in terms of their Enriques diagrams. For linear adjacency a complete answer is obtained. A complete answer is obtained.
(joint work with Joaquim Ro\`e)

Enrico Carlini

Varieties of Simultaneous Sums of Powers for binary forms

The problem of simultaneous decomposition of binary forms as sums of powers of linear forms is studied. For generic forms the minimal number of linear forms needed is found and the space parametrizing all the possible decompositions is described. These results are applied to the study of rational curves.

Marta Casanellas

Non-connected Buchsbaum curves and the Lazarsfeld-Rao property

It is well known that the only non-connected arithmetically Buchsbaum curve in P3 is the disjoint union of two skew lines. The aim of this talk is to give a characterization of non-connected arithmetically Buchsbaum schemes in Pn generalizing this result. We will also use this characterization to discuss a possible generalization of the Lazarsfeld-Rao property for Gorenstein Liaison theory in arbitrary codimension.

Karen A. Chandler

Progress in plump point postulation

Geramita has raised the rich question of: what are the possible Hilbert functions of fat point schemes in projective space? We shall discuss recent constructions and results in this area.

A starting ground is to seek how such a Hilbert function may be prevented from achieving maximal rank. We work from the basic principle that any such hindrance is ``due to'' a visible geometric obstruction, i.e., a positive-dimensional base locus induced by the given postulation. This concept was formally introduced by Harbourne and Hirschowitz, and explored prosperously by Alexander and Hirschowitz; Harbourne; Ciliberto and Miranda; Catalisano, Ellia, and Gimigliano; Trung and Valla; myself, and many others.

We focus to the study of such a Hilbert function in low degree, where the ``worst'' deviations from maximal rank occur. Particularly, we consider conjectures of Iarrobino, viewed as counting the expected linear obstructions to a fat point scheme.

We describe results and techniques, particularly toward verifying Iarrobino's conjectured upper bound on the Hilbert function of a fat point scheme. Further, we exhibit how the techniques generalise (or specialise!) to results on conjectures of Catalisano and Gimigliano in the situation of points on a rational normal curve, as a lower bound for points in linearly general position.

Luca Chiantini

On the postulation of nodes of projection of curves

It is classically known that the geometry of linear series on a smooth curve C Pr is reflected in the postulation of the set N of nodes of a general projection C P2. It turns out, however, that even some extrinsic invariants of the embedding of C in Pr imply restrictions on the Hilbert function of N.

In this talk, mainly the case C P3 is considered. Using a lemma on the growth of non complete linear series on the plane, one can determine bounds on the Hilbert function of N with respect (for instance) to the minimal s such that h0(IC,P3(s)) 0.

Aldo Conca

Cubic forms and Koszul algebras

A standard graded K-algebra R is Koszul if K has a linear resolution as an R-module. The algebra R is said to be quadratic if its defining ideal I is generated by polynomials of degree 2 and it is said to be G-quadratic if I has a Gröbner basis of quadrics in some coordinate system and with respect to some term order. It is well known that:
G-quadratic Koszul quadratic
and that these implications are strict in general. These properties appear naturally in various contexts and their study have attracted the attention of many researchers in the last three decades. For instance, many classical varieties (Grassmannians, Schubert varieties, flag manifolds,...) are Koszul and even G-quadratic in their natural embedding and any projective variety can be embedded in such a way that it is G-quadratic.

In the talk we will address the problem of deciding whether an Artinian Gorenstein algebras with socle in degree 3 is quadratic, Koszul or G-quadratic. These algebras are in bijective correspondence with cubic forms via apolarity, a duality which is known also as Macaulay inverse system.

The main results are:


Theorem 1 For a generic cubic f Pn the dual ring Af is Koszul and not G-quadraic. For a generic singular cubic f Pn the dual ring Af is G-quadraic.


Theorem 2 For a cubic form f Pn with n=2,3 one has:

a) the dual ring Af is Koszul iff Af is quadratic iff f has no polar quadric of rank 1.

b) Af is G-quadratic iff f is singular and f has no polar quadric of rank 1.


For n > 3 we do not know whether quadraticity of Rf implies already its Koszulness and whether the quadraticity of Rf can be characterized in terms of ranks of polar quadrics.

The main technique employed is based on the notion of Koszul filtration and of Gröbner flag.

Alessandro Gimigliano

0-dimensional schemes, tensor rank and secant varieties

To give a stratification by rank of tensors of given format is equivalent to study the dimension of higher secant varieties of Segre varieties. Via Terracini's Lemma, or apolarity, this can problem can be studied on the Hilbert funcion of 2-fat points in products of Projective spaces. The same approach can be used for special tensors (symmetric, skew-symmetric, partially simmetric), by studying fat points on varieties such as Veronesean, Grassmannians and others.

Sudhir Ghorpade

Finite point set configurations and torus actions on Grassmannians

Consider the moduli space of Y = Y(m,n) of n-point configurations in the m-1-dimensional projective space over a field of characteristic zero. This is closely related to certain torus actions on Grassmannians. Algebraically, the space Y corresponds to a certain ring Q of invariants. We shall discuss the difficulties in understanding the combinatorics of the graded ring Q and in particular, in explicitly obtaining its Hilbert function. We, then, describe an explicit formula for the Hilbert function in a special case. Connections with representations of general linear groups will also be discussed.
(This is a joint work with D.-N. Verma).

Elena Guardo

Fat Points Schemes on a Smooth Quadric

We study 0-dimensional fat points schemes on a smooth quadric Q @ P1×P1 , and we characterize those schemes which are arithmetically Cohen-Macaulay (aCM for short) as subschemes of Q giving their Hilbert matrix and bigraded Betti numbers.
In particular, we can compute the Hilbert matrix and the bigraded Betti numbers for fat points schemes with homogeneous multiplicities and whose support is a complete intersection (C.I. for short).
Moreover, we find a minimal set of generators for schemes of double points whose support is aCM.

Anthony Iarrobino

Some Gorenstein Artinian algebras of height four (dvi)

This talk reports on joint work with H. Srinivasan. We first give some context.

F. H. S. Macaulay termed Gorenstein Artinian algebras, as they are now called, ``principal systems'' because their inverse system - dualizing module - has a single generator. They are the simplest class of Artinian algebras extending the complete intersections. It was Grothendieck who named their analogues in arbitrary codimension ``Gorenstein'' algebras, a name that stuck after H. Bass's fundamental article, ``The ubiquity of Gorenstein rings'' exploring their homological aspects. In codimension two, these are complete intersections. In codimension three D. Buchsbaum and D. Eisenbud proved the seminal Pfaffian structure theorem that has abetted their exploration. But in codimension four there is no general structure theorem; not even the set of possible Hilbert functions is known.

In this talk we first describe some of the recent history of the study of graded Gorenstein algebras, especially the Artinian ones, in codimension three and four. We assume that the Gorenstein Artinian [GA] algebra A=R/I, R=k[x1,,xr] is graded, and has Hilbert function T of a fixed socle degree j. Using the generator of the inverse system of A, one may naturally parametrize the family PGOR(T) of all such graded algebras as a quasiprojective subscheme of a projective space \mathbb PN, N=((r+j-1) || j)-1.

D. Buchsbaum and D. Eisenbud, R. Stanley, S. J. Diesel, T. Harima, J. O. Kleppe, M. Boij, G. Valla and A. Conca and others studied height three graded GA algebras, determining the Hilbert functions T (``Gorenstein sequences'') and graded Betti numbers B that may occur. Susan Diesel showed that PGOR(T) is an irreducible variety, and determined that the closure of the Betti stratum PGOR(T,B) is a union of the Betti strata PGOR(T,B) for B B ; then J. O. Kleppe showed that PGOR(T) is even smooth in codimension three. M. Boij determined the dimension of PGOR(T,B), and together with resuls of A. Conca and G. Valla, this gives a second proof of the smoothness of PGOR(T). The analogous lifting problem asks which arithmetically Gorenstein sets of points, reduced curves, or dimension d smooth varieties may have a given h-vector T: this been studied by J. Migliore and A. Geramita, G. Valla, with N. V. Trung and J. Herzog, A. Ragusa and G. Zappalá, and others.

In codimension four, M. Boij gave the first example of a PGOR(T) that has several irreducible components. T. Harima, also A. Geramita, A. Harima with Y. Shin, and J. Migliore and U. Nagel, also Y. Cho and the speaker, V. Kanev and the speaker have studied GA algebras of given Hilbert functions constructed from sets of points in \mathbb Pn, or from zero-dimensional schemes of \mathbb Pn. A. Geramita et al, then J. Migliore et al constructed GA algebras having extremal Betti numbers among the those satisfying a weak Lefschetz condition: A Geramita et al used linking in complete intersections, which applies to many but not all Gorenstein sequences T satisfying D(T) j/2 is an O-sequeence; J. Migliore et al used linking in Gorenstein ideals, which applied to all such T. Results to 1999 were surveyed by V. Kanev and the speaker in SLN # 1721.

We then report on a joint study with H. Srinivasan of GA algebras A of Hilbert function T=(1,4,7,). First, we give a structure theorem for A=R/I such that I2 @ wx,wy,wz, and we study those with I2 @ w2,wx,wy. The former set often determines an irreducible component of PGOR(T), and the latter set, presumably in the closure of the first, has been an intriguing puzzle.

Recall that the set of Gorenstein sequences T=(1,4,,1) in height four is unknown. It is not even known whether such sequences must be unimodal - increasing until a maximum value is attained, then constant until degree j/2 , then decreasing.

Theorem 1 Every Gorenstein sequence T=(1,4,a,,1) of socle degree j with a 7 satisfies the condition, DT j/2 is an O-sequence.

This condition is stronger than being unimodal. The proof, however, suggests the contrary conjecture that there are Gorenstein sequences T=(1,4,10,) for which DT j/2 is not an O-sequence!

Finally, in joint work with H. Srinivasan still in progress, we determine the possible Betti strata and irreducible components of the scheme PGOR(Th), Th=(1,4,7,h,7,4,1). When h=8,9,10, the family PGOR(Hh) has several irreducible components, which is in contrast to the known irreducibility, even smoothness of PGOR(H) in embedding dimension three. The Betti strata here are sparse, and have much to do with the Hilbert schemes of curves in \mathbb P3. The proofs use properties of minimal resolutions, the Hilbert schemes of curves in \mathbb P3, many of the above-mentioned results about height three Gorenstein algebras, and also the Gotzmann Hilbert scheme theorems.

Martin Kreuzer

Computational Aspects of Zero-Dimensional Schemes

In this talk we examine several questions arising in Computational Commutative Algebra which are motivated by the study of zero-dimensional subschemes of projective spaces.


The first problem is the efficient computation of the vanishing ideal of a zero-dimensional scheme. We discuss several methods based on the Buchberger-Möller algorithm and apply the results in interpolation theory and statistics.


The second problem is the task of efficiently checking various uniformity conditions for zero-dimensional schemes. Some partial answers based on the theory of canonical modules are given together with applications in coding theory.


The third question is how one can compute the minimal graded free resolution of the vanishing ideal efficiently, in particular for the ideal of a generically chosen set of points. We discuss the status of the minimal resolution conjecture both from the computational and the theoretical point of view.


Finally, we mention the relation between Hilbert functions of fat points and linear systems of divisors on rational surfaces, as well as some differential objects associated to zero-dimensional schemes.

Antonio Laface

Zero dimensional schemes and special linear systems

Consider a non special linear system L = | H-mipi | , given on a smooth algebraic surface S and H is a very ample non special divisor on it. Suppose that for a generic p S, L- 2p is aspecial sys tem, in this case we call L a "pre-special system". In thistalk we give a characterization of such systems

Mustapha Lahyane

Irreducibility of (-1)-Divisors on Smooth Rational Surfaces

The aim is to give a characterization for a (-1)-divisor on anticanonical rational surface to be irreducible.

Rosa Maria Miro' Roig

K3 surface: moduli spaces of vector bundles and Hilbert schemes of 0-dimensional subschemes.

Let X be a K3 surface and H an ample divisor on X. Roughly speaking the goal of this talk is to relate the geometry of moduli spaces MX,H(r;c1,c2) of H-stable vector bundles on X to the geometry of Hilbert schemes, Hilbl(X), of 0-dimensional subschemes of X of lenght l. In particular, I will focus my attention on the problem of determining invariants (r,c1,c2,l) for wich the moduli space MX,H(r;c1,c2) and the punctual Hilbert scheme Hilbl(X) are birational.

Manizheh Nafari

Gelfand-Kirillov Dimension

To be announced

Uwe Nagel

A property of points in uniform position

The interest in sets of points with the uniform position property is to a great deal motivated by the fact that over a field of characteristic zero the general heyperplan section of an integral curve has this property. In the talk we will discuss a property of the minimal free resolution of points in uniform position. This leads to further questions. We indicate how answers to these questions could help to lift information from the general hyperplane section of a curve to the curve itself.

Oleg Bogoyavlenskij

Infinite-dimensional Lie groups of symmetries of the ideal MHD equilibrium equations

Infinite-dimensional abelian Lie groups of symmetries Gm are discovered for the system of ideal magnetohydrodynamics equilibrium equations. The groups Gm are isomorphic to the direct sums Am Am R+ Z2 Z2 Z2 where Am is the additive Lie group of smooth functions on R3 that are constant on the magnetic surfaces for a given MHD equilibrium and R+ is the multiplicative group of positive numbers. The groups of symmetries Gm have additional structure of modules over the associative algebras Am Am. The Lie groups Gm have applications in the method of symmetry transforms for constructing the ideal MHD equilibria. The new symmetry transforms break the geometrical symmetries of the field-aligned MHD equilibria, depend on the three spatial variables x,y,z and are given by the explicit algebraic formulae.

Claudia Polini

A conjecture on the core of an ideal

This is joint work with Bernd Ulrich. Let R be a local Gorenstein ring with infinite residue field, and let I be an R-ideal. The core of I is defined to be the intersection over all (minimal) reductions of I. This object was first studied by Rees and Sally and later by Huneke and Swanson. In this talk we give a general description of the core of I, that had already been conjectured in our earlier work.

Leslie Roberts

Certain projective curves with unusual Hilbert function

I give an effective algorithm for computing the Hilbert function of projective monomial curves and for deciding if the homogeneous co-ordinate ring of such a curve is Cohen-Macaulay. I will review this algorithm, and then illustrate it with several interesting examples. The algorithm can be used both for specific curves and infinite families of curves.

Joaquim Ro\`e

On specialization of cluster schemes

We shall report on recent results about families of cluster schemes and applications

Sonia Spreafico

A new construction of codimension 3 Gorenstein ideals with applications

I present an algebraic construction of codimension 3 Gorenstein ideals which has a nice geometrical interpretation. Using this construction I study the Gorenstein liaison class of some particular configuration of points in P3 and some particular generalized stick figures.
(Joint work with C. Bocci, G. Dalzotto, R. Notari)

Rosario Strano

Biliaison classes of curves in P3

We characterize the curves in P3 which are minimal in their biliaison class. Such curves are exactly the curves which do no admit an elementary descending biliaison. As a consequence we have that every curve in P3 can be obtained from a minimal one by means of a finite sequence of ascending elementary biliaisons.

Jaydeep V. Chipalkatti

Schemes apolar to a net of ternary quartics

The classical Waring's problem for polynomials is the following: given n-ary forms F1, ..., Fr of degree d, we would like to find linear forms L1, ...,Ls, such that each Fi is expressible as a linear combination of L1d, ..., Lsd. The problem is equivalent to finding zero-dimensional closed subschemes of Pn having length s, which are apolar to the Fi. In general, it is not even known for which values of s such linear forms may be found.

In this talk we will show that if F1, F2, F3 are general ternary quartics, then there are exactly four 9-tuples {L1, ..., L9 } which solve the problem. The calculation involves applying the Grothendieck-Riemann-Roch theorem to a vector bundle on a certain symmetric product of an elliptic curve.

William Travers

The Roller-Coaster Conjecture and Artin Level Algebras

It is a ridiculous simplification to say that much has been done to categorize the h-vectors of commutative algebras. In this talk, I will present a new conjecture about the h-vectors of Artin Level algebras inspired by the combinatorics of cliques in well-covered graphs. Significant evidence exists for the conjecture, but it remains open despite its accessible nature.

Bernd Ulrich

Castelnuovo-Mumford regularity and linkage

This is a report on joint work with Marc Chardin. We prove bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. Some of our estimates work in any characteristic and apply to ideals that are not necessarily saturated or equidimensional. Our methods are based on linkage theory and the notion of F-rational singularities.

Adam Van Tuyl

Fat Points in P1 ×P1 and their Hilbert functions

Let X be a set of fat points in P1 x P1. We show how to associate to X two tuples aX and bX that contain information about the scheme X. Specifically, we show how to use aX and bX to calculate all but a finite number of values of the Hilbert function of X. We also show how to use aX and bX to determine if X is arithmetically Cohen-Macaulay.

Giannina Beccari - Carla Massaza

Realizable sequences for multiple points

To any multiple point in Pn we associate a set of realizable numerical sequences, whose entries are completely determined by the Hilbert function of its local ring. The multiple point can be considered as a limit of a scheme of simple points, having the same set of realizable sequences.

Giuseppe Zappalà

Some constructions of reduced aCM schemes

Given an admissible Hilbert function H for an arithmetically Cohen-Macaulay scheme, what are all the possible graded Betti numbers compatible with H? To approach this problem we give two constructions of reduced aCM schemes: partial intersections in \mathbb Pn and partial Gorenstein in \mathbb P3. For such schemes the first and last graded Betti numbers can be easily computed in terms of their combinatorial support (so in \mathbb P3 we know all graded Betti numbers).
(This is a joint work with Alfio Ragusa).

Marina Zompatori

Homogeneous linear systems of plane curves with a composite number of base points

We extend the range of homogeneous linear systems of plane curves that are known to be non-special. Using a degeneration technique developed by C. Ciliberto and R. Miranda, we obtain a result that implies as corollary Evain's theorem on the non-speciality of homogeneous systems through 4h points and we deduce a similar result for 9k points.

Susan Marie Cooper

Subsets of Complete Intersections in \mathbb P2: Their Hilbert Functions and Associated 2-Type Vectors

We call a set of points \mathbb Y in \mathbb P2 a complete intersection, denoted \mathbb Y = C.I.(a,b), if the ideal of \mathbb Y can be generated by two homogeneous polynomials F and G of degrees a and b, where F and G are necessarily without common factors and 1 a b. It is easy to show that all C.I.(a,b) have the same Hilbert function. Given positive integers a b and a Hilbert function H of a finite set of points in \mathbb P2, we show how to determine if there exists a \mathbb Y = C.I.(a,b) and a subset \mathbb X of \mathbb Y such that the Hilbert function of \mathbb X is H. We also state this result in terms of 2-type vectors, which are in 1-1 correspondence with Hilbert functions of finite sets of points in \mathbb P2. In addition, we will see CoCoA programmes written for related problems. This work is from my M.Sc. thesis. I have similar results for \mathbb P3 and am currently working on analogues for \mathbb Pn as part of my Ph.D. dissertation.




File translated from TEX by TTH, version 3.08.
On 23 May 2002, 17:55.