Tony Iarrobino's deep insights into the structure of Artinian Gorenstein algebras, Hilbert schemes, and the geometry of punctual schemes have left a lasting mark on modern algebraic geometry and commutative algebra. In this talk, I will share a collection of my favorite problems, most of which are still open, that are inspired by his work. These include questions about the properties of...
Nancy Abdallah, Jacques Emsalem and Tony Iarrobino classified net of conics in the paper “Nets of Conics and associated Artinian algebras of length 7”, which is an extended version of the unpublished preprint “Réseaux de coniques et algèbres de longueur sept associées” by Jacques and Tony written back in 1972. In my talk, I will explain how to use this results to determine which smooth Fano...
I will report on joint work with A. Boralevi and M.L. Fania, about linear spaces and quadric
surfaces contained in the Pfaffian hypersurface in P^{14}. I will then explain the connections with congruences of lines in the 5-dimensional projective space and with vector bundles
We introduce the notion of an “initial condition” for a module M over a commutative Noetherian local ring (A,m), allowing for a recursive construction of its “solution modules”. If M has zero-dimensional support, such as the residue field of A, we demonstrate that the solution module E(M) is its “linear closure”, turning out to be an injective hull of M. The construction of E(M) for finitely...
It is an extremely elusive problem to determine which standard Artinian Gorenstein graded K-algebras satisfy the weak Lefschetz property (WLP). Codimension 2 Artinian Gorenstein graded K-algebras have the WLP and it is open to what extent such result might work for codimension 3 Artinian Gorenstein graded K-algebras.
In this talk I will summarize what we know about this problem and, in...
Given a form F of degree d, its tangential decompositions are additive decompositions of F that involve only terms of type L^{d−1}G, where L and G are linear forms. To any such decomposition, we can naturally associate 0-dimensional apolar schemes made of simple points (when L= G, projectively), and 2-jets (when Land Gare not proportional). Among these schemes, it is possible to find...
We present a refined version of the cactus algorithm in the case of local schemes, improving both reliability and efficiency in describing the local structure of Artinian Gorenstein schemes.
We study G-graded Artinian algebras having Poincaré duality and their Lefschetz properties. We prove the equivalence between the toric setup and the G-graded one. We prove a Hessian criterion in the G-graded setup. We provide an application to toric geometry. We propose a question about Jordan types.
(Joint with U. Bruzzo, R. Holanda, W. Montoya.)
If X is a set of reduced points lying on a rational normal curve, the Hilbert function of X is
classically known. Starting from this result, we address the following problem: what are the possible Hilbert functions of a reduced subvariety of a Veronese variety? We provide a general result for any Veronese variety and then derive an effective characterisation of the Hilbert function of points...
We will review some of the work on commuting pairs of matrices that led to the Box Conjecture of Anthony Iarrobino and his collaborators. Then, we will sketch a proof of the Conjecture. The proof hinges naturally on the Burge correspondence between the set of all partitions and a set of binary words. For connection with the algebraic and geometric setup of matrices and nilpotent orbits we use...
Tensor decomposition is a challenging problem from a computational and numerical point of view. This can be explained by the complex and still unrevealed geometry hidden behind the scene.
We will explore this complex geometry via the lens of algebra, investigating how Artinian algebras can be naturally associated to general (additive) decompositions of tensors. We will review algebraic...
The Jordan type of a nilpotent matrix in the dense orbit of the nilpotent commutator of a given nilpotent matrix of Jordan type P is stable, which means that the parts differ by at least two. Fixing a matrix J of stable Jordan type Q, there is an affine space of nilpotent matrices commuting with J.
In recent joint work with A. Iarrobino and L. Khatami, we use some tropical calculations to...
What are the equations for the Hilbert schemes in their natural embeddings? Different set of (high degree) equations have been found. Among them, the determinantal equations by Tony Iarrobino and Steve Kleiman.
In this talk, a new construction of the Hilbert scheme will be given, from which we derive a set of equations of degree one and two.
Our construction extends the description by...
Given a field k and a graded k-algebra A, let |FΨ^h_A and
|HΨ^h_A, be the schemes parameterizing filtered quotients and graded
quotients of A with Hilbert function h. Let |FΨ^{h,t}_A and |HΨ^{h,t}_A
be their subschemes of Artinian quotients of socle type t.
In 1984, Iarrobino proved that, if k is infinite, if A is a
polynomial ring, if t is permissible in a certain sense, and
if h =...
Some 30 years ago, Peter Orlik and Hiro Terao introduced a commutative analog of the Orlik-Solomon (OS) algebra. The OS algebra is the cohomology ring of a hyperplane arrangement complement, and is a quotient of an exterior algebra by a combinatorially determined ideal. The Orlik-Terao (OT) algebra and Artinian version (AOT) have subsequently been studied by many authors (sometimes under the...
We will discuss how Iarrobino’s symmetric decomposition became a central numeric invariant for Gorenstein algebras and discuss the current state of the art: open questions and results related to completed quadrics and Iarrobino’s scheme.
In this talk we recall Saito’s theory about the notions mentioned in the title for an hypersurface. We shall focus more specifically the notion of residue and prove duality statements for the module that they form. Concerning applications we gave the final step of a proof of Saito’s conjecture about the characterisation of singularities which are normal crossings in codimension 2. We mention...
Let (A,m) be a complete local ring and G= gr_{m}(A) its associated graded ring. The problem of the descent of a property from G to A was extensively studied and the answers are predominantly positive. The problems arise passing from A to G, because we may lose many good geometric and algebraic properties. We present an overview up to a recent work with A. De Stefani and M. Varbaro. We...
In the talk we classify irreducible components of Hilbert schemes of 9 and 10 points in affine spaces of any dimension. The main tool is the connection between Hilbert schemes of points and varieties of commuting matrices.
This is joint work with Maciej Galazka and Hanieh Keneshlou.
We recall the connection of the Gorenstein loci in the Hilbert scheme of points on a projective variety X with the cactus varieties of X in some embedding. We give an analogous description of cactus varieties of pencils (which correspond to simultaneous rank). We conclude by showing some examples of cactus varieties of pencils which are equal to their corresponding secant varieties.
This is...
The point process limit of the random matrix spectra are defined via the limit joint densities, relying on some algebraic structures. I will describe some relations between these dynamics and the dynamics of root sets of some random polynomials under iterated application of differential operations on the polynomials.
