Jordan Types of Artinian Algebras and Geometry of Punctual Hilbert Schemes (JTAG)

Liste des Contributions

1. My favorite problems inspired by Tony Iarrobino's work

Prof. Alexandra Seceleanu (University of Nebraska - Lincoln)

23/06/2025 09:00

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...

2. K-stability of nets of conics

Prof. Ivan Cheltsov (University of Edinburgh)

23/06/2025 10:30

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conics in the paper “Nets of Conics and associated Artinian algebras of length 7”, which is an extended version of the unpublished preprint “Réseaux de coniques et algèbres de longueur sept associées” by Jacques and Tony written back in 1972.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...

9. Families of symmetric and skew-symmetric matrices and vector bundles

Prof. Emilia Mezzetti (Università degli Studi di Trieste)

23/06/2025 11:00

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

4. On the definition of a general element and polarization

Prof. Junzo Watanabe (Tokai University)

23/06/2025 15:00

5. Solving Linear Equations

Prof. I-Chiau Huang (Institute of Mathematics, Academia Sinica)

23/06/2025 15:30

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...

23. Problem Session

23/06/2025 16:30

Any open problem, participants would like to share with others is welcomed.

24. Free discussions

23/06/2025 17:00

6. The weak Lefschetz property for Artinian Gorenstein algebras

Prof. Rosa Maria Miró-Roig (Universitat de Barcelona)

24/06/2025 09:00

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...

7. Regularity of tangential decompositions

Dr Daniele Taufer (KU Leuven)

24/06/2025 10:30

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...

8. A refinement of the local cactus algorithm

Prof. Alessandra Bernardi (Università di Trento)

24/06/2025 11:00

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.

3. Cox-Gorenstein Algebras

Prof. Rodrigo Gondim (Universidade Federal Rural de Pernambuco)

24/06/2025 15:00

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.)

10. Hilbert functions of Veronesean subvarieties and Complete Intersections

Dr Stefano Canino (Universytet Warszawski)

24/06/2025 16:30

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...

25. Free discussions

24/06/2025 17:00

11. A proof of the Box Conjecture for commuting pairs of matrices

Prof. Tomaž Košir (University of Ljubljana)

25/06/2025 09:00

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...

12. Tensor decomposition via the analysis of Artinian algebras

Bernard Mourrain (INRIA - Université Côte d'Azur)

25/06/2025 10:30

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...

13. Jordan type stratification of spaces of commuting nilpotent matrices

Prof. Mats Boij (KTH Royal Institute of Technology)

25/06/2025 11:00

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...

14. Low degree equations for the Hilbert Schemes

Prof. Laurent Evain (Université d’Angers)

26/06/2025 09:00

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...

15. Maximal Hilbert functions of Artinian quotients of a product ring

Prof. Jan O. Kleppe (Oslo Metropolitan University)

26/06/2025 10:30

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 =...

16. Lefschetz properties of the Artinian Orlik-Terao algebra

Prof. Hal Schenck (Auburn University)

26/06/2025 11:00

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...

17. Iarrobino’s symmetric decomposition: open questions and some developments

Prof. Joachim Jelisiejew (Uniwersytet Warszawski)

26/06/2025 15:00

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.

18. Around logarithmic forms and vector fields, with a view on their modules of residues

Prof. Michel Granger (Université d’Angers)

26/06/2025 15:30

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...

19. Results and open problems passing from a local ring to its associated graded ring

Prof. Maria Evelina Rossi (Università degli Studi di Genova)

26/06/2025 16:30

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...

26. Free discussions

26/06/2025 17:00

20. Irreducible components of Hilbert schemes of points

Prof. Klemen Šivic (University of Ljubljana)

27/06/2025 09:00

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.

21. Irreducible components of Hilbert schemes and application to secant varieties

Dr Maciej Galazka (Daegu-Gyeongbuk Institute of Science and Technology)

27/06/2025 10:30

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...

22. Dynamics of eigenvalues of random matrices under iterated transforms

Prof. André Galligo (Université Côte d’Azur)

27/06/2025 11:00

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.

27. Problem Session

27/06/2025 15:00

28. Problème Session

27/06/2025 16:30

29. Free discussions

27/06/2025 17:00