Enriques diagrams and adjacency of planar curve singularities
Varieties of Simultaneous Sums of Powers for binary forms
Nonconnected Buchsbaum curves and the LazarsfeldRao property
Progress in plump point postulation
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 positivedimensional 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.
On the postulation of nodes of projection of curves
It is classically known that the geometry of linear series on a smooth curve C Ì P^{r} is reflected in the postulation of the set N of nodes of a general projection C® P^{2}. It turns out, however, that even some extrinsic invariants of the embedding of C in P^{r} imply restrictions on the Hilbert function of N.
In this talk, mainly the case C Ì P^{3} 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 h^{0}(I_{C,P3}(s)) ¹ 0.
Cubic forms and Koszul algebras
A standard graded Kalgebra R is Koszul if K has a linear resolution
as an Rmodule. 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
Gquadratic if I has a
Gröbner basis of quadrics in some coordinate system and with
respect to some term
order. It is well known
that:

In the talk we will address the problem of deciding whether an Artinian Gorenstein algebras with socle in degree 3 is quadratic, Koszul or Gquadratic. 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 Î P^{n} the
dual ring A_{f}
is Koszul and not Gquadraic. For a generic singular cubic f Î P^{n} the dual
ring A_{f} is Gquadraic.
Theorem 2 For a cubic form f Î P^{n} with
n=2,3 one has:
a) the dual ring A_{f} is Koszul iff A_{f} is quadratic iff f has no polar quadric of rank 1.
b) A_{f} is Gquadratic iff f is singular and f has no polar quadric of rank 1.
For n > 3 we do not know whether quadraticity of R_{f}
implies already its Koszulness and whether the quadraticity of R_{f} 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.
0dimensional schemes, tensor rank and secant varieties
Finite point set configurations and torus actions on Grassmannians
Consider the moduli space of Y = Y(m,n) of npoint
configurations in the m1dimensional 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).
Fat Points Schemes on a Smooth Quadric
We study 0dimensional fat points schemes on a smooth quadric
Q @ P^{1}×P^{1} , and we characterize those
schemes which are arithmetically CohenMacaulay (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.
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[x_{1},¼,x_{r}] 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 P^{N}, N=((r+j1)  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¢ ³ 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 hvector 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 P^{n}, or from zerodimensional schemes of \mathbb P^{n}. 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 Osequeence; 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=(1,4,7,¼). First, we give a structure theorem for A=R/I such that I_{2} @ áwx,wy,wzñ, and we study those with I_{2} @ áw^{2},wx,wyñ. 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=(1,4,¼,1) 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 éj/2 ù, then decreasing.
Theorem 1 Every Gorenstein sequence T=(1,4,a,¼,1) of socle degree j with a £ 7 satisfies the condition, DT_{ £ j/2} is an Osequence.
This condition is stronger than being unimodal. The proof, however, suggests the contrary conjecture that there are Gorenstein sequences T=(1,4,10,¼) for which DT_{ £ j/2} is not an Osequence!
Finally, in joint work with H. Srinivasan still in progress, we determine the possible Betti strata and irreducible components of the scheme PGOR(T_{h}), T_{h}=(1,4,7,h,7,4,1). When h=8,9,10, the family PGOR(H_{h}) 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 P^{3}. The proofs use properties of minimal resolutions, the Hilbert schemes of curves in \mathbb P^{3}, many of the abovementioned results about height three Gorenstein algebras, and also the Gotzmann Hilbert scheme theorems.
Computational Aspects of ZeroDimensional Schemes
In this talk we examine several questions arising in Computational Commutative Algebra which are motivated by the study of zerodimensional subschemes of projective spaces.
The first problem is the efficient
computation of the vanishing ideal of a zerodimensional
scheme. We discuss several methods based on the BuchbergerMöller
algorithm and apply the results in interpolation
theory and statistics.
The second problem is the task of efficiently checking
various uniformity conditions for zerodimensional 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 zerodimensional schemes.
Zero dimensional schemes and special linear systems
Consider a non special linear system L =  Håm_{i}p_{i}  , given on a smooth algebraic surface S and H is a very ample non special divisor on it. Suppose that for a generic p Î S, L 2p is aspecial sys tem, in this case we call L a "prespecial system". In thistalk we give a characterization of such systems
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.
K3 surface: moduli spaces of vector bundles and Hilbert schemes of 0dimensional 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 M_{X,H}(r;c_{1},c_{2}) of Hstable vector bundles on X to the geometry of Hilbert schemes, Hilb^{l}(X), of 0dimensional subschemes of X of lenght l. In particular, I will focus my attention on the problem of determining invariants (r,c_{1},c_{2},l) for wich the moduli space M_{X,H}(r;c_{1},c_{2}) and the punctual Hilbert scheme Hilb^{l}(X) are birational.
GelfandKirillov Dimension
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.
Infinitedimensional Lie groups of symmetries of the ideal MHD equilibrium equations
Infinitedimensional abelian Lie groups of symmetries G_{m} are discovered for the system of ideal magnetohydrodynamics equilibrium equations. The groups G_{m} are isomorphic to the direct sums A_{m} ÅA_{m} ÅR^{+} ÅZ_{2} ÅZ_{2} ÅZ_{2} where A_{m} is the additive Lie group of smooth functions on R^{3} 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 G_{m} have additional structure of modules over the associative algebras A_{m} ÅA_{m}. The Lie groups G_{m} have applications in the method of symmetry transforms for constructing the ideal MHD equilibria. The new symmetry transforms break the geometrical symmetries of the fieldaligned MHD equilibria, depend on the three spatial variables x,y,z and are given by the explicit algebraic formulae.
A conjecture on the core of an ideal
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 coordinate ring of such a curve is CohenMacaulay. 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.
On specialization of cluster schemes
We shall report on recent results about families of cluster schemes and applications
A new construction of codimension 3 Gorenstein ideals with applications
Biliaison classes of curves in P^{3}
Schemes apolar to a net of ternary quartics
In this talk we will show that if F_{1}, F_{2}, F_{3} are general ternary quartics, then there are exactly four 9tuples {L_{1}, ..., L_{9} } which solve the problem. The calculation involves applying the GrothendieckRiemannRoch theorem to a vector bundle on a certain symmetric product of an elliptic curve.
The RollerCoaster Conjecture and Artin Level Algebras
CastelnuovoMumford regularity and linkage
Fat Points in P^{1} ×P^{1} and their Hilbert functions
Realizable sequences for multiple points
To any multiple point in P^{n} 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.
Some constructions of reduced aCM schemes
Homogeneous linear systems of plane curves with a composite number of base points
Subsets of Complete Intersections in \mathbb P^{2}: Their Hilbert Functions and Associated 2Type Vectors
We call a set of points \mathbb Y in \mathbb P^{2} 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 P^{2}, 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 2type vectors, which are in 11 correspondence with Hilbert functions of finite sets of points in \mathbb P^{2}. 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 P^{3} and am currently working on analogues for \mathbb P^{n} as part of my Ph.D. dissertation.