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

23 juin 2025, 07:00 → 27 juin 2025, 17:50 Europe/Paris

Salle de conférence (Nice)

The purpose of the conference is to celebrate the work of Anthony Iarrobino in algebra and geometry. The theme of the conference focuses on two threads: the Jordan type of Artinian algebras and the geometry of Hilbert schemes. Artinian algebras, characterized by their finite-dimensional structure, play a crucial role in various areas of mathematics, including representation theory, commutative algebra, and algebraic geometry. The Jordan type, associated with the Jordan canonical form, provides insights into the structure of these algebras. 

The punctual Hilbert scheme, which parametrizes zero-dimensional subschemes of a fixed length in an algebraic variety, is a fundamental object in algebraic geometry with deep connections to combinatorics, singularity theory, and representation theory. Understanding the geometry of these schemes, particularly through the lens of Jordan types, opens new avenues for exploring the intricate relationship between algebra and geometry.

The conference aims to bring together leading researchers and young scholars to discuss recent advancements, share methodologies, and foster collaboration in this specialized but highly impactful area of mathematics. Topics will include the classification of Jordan types in various contexts, the role of Artinian algebras in deformation theory, and the geometric properties of punctual Hilbert schemes. By examining these themes, the conference seeks to deepen the understanding of the algebra-geometry interface and its implications for broader mathematical research.

lun. 23 juin

08:30 → 08:55 Registration

09:00 → 09:50 My favorite problems inspired by Tony Iarrobino's work

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 Artinian ring and the homological and asymptotic behavior of punctual schemes and related Macaulay inverse systems. Along the way, I’ll highlight the powerful techniques developed or influenced by Tony and discuss how his vision continues to inspire new directions in the field.

Orateur: Prof. Alexandra Seceleanu (University of Nebraska - Lincoln)

Seceleanu.pdf

10:00 → 10:25 Coffee Break

10:30 → 10:50 K-stability of nets of conics

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 3-folds in the deformation family No. 2.24 (divisors of degree (1,2) in the product of two projective planes, see https://www.fanography.info/) admit Kahler-Einstein metric and which does not.

Orateur: Prof. Ivan Cheltsov (University of Edinburgh)

Cheltsov.pdf

11:00 → 11:50 Families of symmetric and skew-symmetric matrices and vector bundles

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

Orateur: Prof. Emilia Mezzetti (Università degli Studi di Trieste)

Mezzetti.pdf

12:25 → 14:55 Lunch

15 to 20mn walk from Laboratoire J.A. Dieudonné to the place of the lunch

15:00 → 15:20 On the definition of a general element and polarization

Orateur: Prof. Junzo Watanabe (Tokai University)

Watanabe.pdf

15:30 → 15:50 Solving Linear Equations

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 generated M, hence injective hull of M, is explicit and computable, devoid of the need for Zorn’s lemma. As an application, we improve Baer’s criterion for a module N with zero-dimensional support to be injective: If any A-homomorphism from m to N lifts to A, then N is injective. For the case that A is Artinian and M is its residue field, we demonstrate how systems of linear equations are explicitly solved to obtain an injective hull.

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

Huang.pdf

16:00 → 16:25 Coffee break

16:30 → 17:00 Problem Session

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

17:00 → 17:50 Free discussions

mar. 24 juin

09:00 → 09:50 The weak Lefschetz property for Artinian Gorenstein algebras

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 particular, I will show that all Artinian Gorenstein K-algebras of codimension 3 with arbitrary socle degree d, arbitrary Sperner number α and at least three peaks (i.e., three values of the h-vector reaching the Sperner number) do satisfy the WLP.

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

Miro-Roig.pdf

10:00 → 10:25 Coffee break

10:30 → 10:50 Regularity of tangential decompositions

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 irredundant ones (i.e. those minimal by inclusion) that are not regular in degree d. Nonetheless, this never happens for the shortest schemes (i.e. those minimal by length), as we can always “refine” such an irregular scheme to obtain a d-regular and strictly shorter one, of the same type. In this talk, I will illustrate this construction and briefly discuss its implications for the explicit computation of minimal tangential decompositions of forms. This is based on a joint work with A. Bernardi and A. Oneto.

Orateur: Dr Daniele Taufer (KU Leuven)

Taufer.pdf

11:00 → 11:50 A refinement of the local cactus algorithm

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.

Orateur: Prof. Alessandra Bernardi (Università di Trento)

Bernardi.pdf

12:25 → 14:55 Lunch

15 to 20mn walk from Laboratoire J.A. Dieudonné to the place of the lunch

15:00 → 15:50 Cox-Gorenstein Algebras

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.)

Orateur: Prof. Rodrigo Gondim (Universidade Federal Rural de Pernambuco)

Gondim.pdf

16:00 → 16:25 Coffee break

16:30 → 16:50 Hilbert functions of Veronesean subvarieties and Complete Intersections

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 lying on a Veronese surface. As an application, we completely classify the complete intersections of the ambient space that lie on a Veronese surface, and we formulate a conjecture for Veronese varieties in higher dimensions.
This is joint work with Prof. Enrico Carlini.

Orateur: Dr Stefano Canino (Universytet Warszawski)

Canino.pdf

17:00 → 17:50 Free discussions

mer. 25 juin

09:00 → 09:50 A proof of the Box Conjecture for commuting pairs of matrices

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 Shayman’s results on invariant subspaces of a nilpotent matrix.
This is joint work with John Irving and Mitja Mastnak.

Orateur: Prof. Tomaž Košir (University of Ljubljana)

Kosir.pdf

10:00 → 10:25 Coffee break

10:30 → 10:50 Tensor decomposition via the analysis of Artinian algebras

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 geometric approaches, which reduces tensor decomposition to direct eigenvalue and eigenvector computations when the rank is small.
When the rank is higher, the geometry is much more complex. We will see how the tensor decomposition problem can be transformed into a simultaneous diagonalization problem of extended tensors or extensor. We will connect the varieties of extensors with the punctual Hilbert scheme, via families of commuting matrices and illustrate the concepts and the computational approach on some examples.

Orateur: Bernard Mourrain (INRIA - Université Côte d'Azur)

Mourrain.pdf

11:00 → 11:50 Jordan type stratification of spaces of commuting nilpotent matrices

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 determine equations defining the loci of each partition P for which Q is the generic commuting partition. We also propose a conjecture generalizing this result to arbitrary stable Q. A key ingredient is the recent proof of the Box Conjecture by J. Irving, T. Koˇsir and M. Mastnak.

Orateur: Prof. Mats Boij (KTH Royal Institute of Technology)

Boij.pdf

jeu. 26 juin

09:00 → 09:50 Low degree equations for the Hilbert Schemes

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 Nakajima of Hilb^{p}(A^{2}) to the projective case and a non constant polynomial. The Hilbert scheme is described as a quotient of a scheme of quiver representations.
The equations are reminiscent of the Plücker relations for Grassmannians: they are explicit and built formally with permutations on indexes on the Plücker coordinates.

Orateur: Prof. Laurent Evain (Université d’Angers)

Evain.pdf

10:00 → 10:25 Coffee break

10:30 → 10:50 Maximal Hilbert functions of Artinian quotients of a product ring

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 = h^I where
h^I(p) := min{ a(p), sum{q>0} t(q)a(q − p) } and a(i) := dim A_i,
then |FΨ^{h,t}_A is an affine space bundle over |HΨ^{h,t}_A, and
|HΨ^{h,t}_A is nonempty, irreducible and covered by open subschemes,
each isomorphic to /A^N with N explicit. For any A, there's a similar
maximal h, but it's not necessarily equal to h^I.
In this talk, we analyze the case where A := SxT and h \neq h^I. When
S := k[x], a polynomial ring in one variable, we prove that |FΨ^{h,t}_A
and |HΨ^{h,t}_A are close to be as nice as when h = h^I. In 2001, Cho and
Iarrobino gave such examples with T := k[y,z]/(z^5) in the graded case. The
new work described here is joint work with Steve Kleiman.

Orateur: Prof. Jan O. Kleppe (Oslo Metropolitan University)

Kleppe.pdf

11:00 → 11:50 Lefschetz properties of the Artinian Orlik-Terao algebra

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 guise as the “algebra of reciprocal forms”). It has surprising connections to classical algebraic geometry (for example, to certain blowups of projective space). We analyze Lefschetz properties of the Artinian Orlik-Terao algebra.

Orateur: Prof. Hal Schenck (Auburn University)

Schenck.pdf

12:25 → 14:55 Lunch

15 to 20mn walk from Laboratoire J.A. Dieudonné to the place of the lunch

15:00 → 15:20 Iarrobino’s symmetric decomposition: open questions and some developments

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.

Orateur: Prof. Joachim Jelisiejew (Uniwersytet Warszawski)

Jelisiejew.pdf

15:30 → 15:50 Around logarithmic forms and vector fields, with a view on their modules of residues

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 also a characterisation of singularities with Gorenstein singular locus. Depending on time left we shall develop more examples of residue modules, for curves, hyperplane arrangement and/or sketch a generalisation of the theory to singularities in higher codimension.

Orateur: Prof. Michel Granger (Université d’Angers)

Granger.pdf

16:00 → 16:25 Coffee break

16:30 → 16:50 Results and open problems passing from a local ring to its associated graded ring

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 introduce a homogenization technique which allows to relate Gto the special fiber and A to the generic fiber of a “Groebner-like” deformation. Using this technique we prove sharp results concerning the connectedness of A and G.

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

Rossi.pdf

17:00 → 17:50 Free discussions

ven. 27 juin

09:00 → 09:50 Irreducible components of Hilbert schemes of points

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.

Orateur: Prof. Klemen Šivic (University of Ljubljana)

Sivic.pdf

10:00 → 10:25 Coffee break

10:30 → 10:50 Irreducible components of Hilbert schemes and application to secant varieties

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 a complementary talk to Klemen Sivic's presentation, and is based on joint work with Hanieh Keneshlou and Klemen Sivic.

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

Galazka.pdf

11:00 → 11:50 Dynamics of eigenvalues of random matrices under iterated transforms

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.

Orateur: Prof. André Galligo (Université Côte d’Azur)

Galligo.pdf

12:25 → 14:55 Lunch

15 to 20mn walk from Laboratoire J.A. Dieudonné to the place of the lunch

15:00 → 15:50 Problem Session

16:00 → 16:25 Coffee break

16:30 → 17:00 Problème Session

17:00 → 17:50 Free discussions