# Main content

## Top content

### Program

#### Speakers

Alessio Caminata – Symmetric signature

Alexandru Constantinescu – Linear syzygies and regularity

Alessio D’Alì – Deviations of graded algebras

Emanuela De Negri – A Gorenstein simplicial complex for symmmetric minors

Gunnar Fløystad – Letterplace and co-letterplace ideals of posets

Takayuki Hibi – Gorenstein Fano polytopes and quadratic Gröbner bases

June Huh – Hodge theory for combinatorial geometries

Martina Juhnke-Kubitzke – Gap vectors of real projective varieties

Thomas Kahle – Plethysm and lattice point counting

Lukas Katthän – The Golod property of Stanley-Reisner rings

Michal Lason – On the toric ideal of a matroid and related combinatorial problems

Dinh Van Le – On h-vectors of broken circuit complexes

Antonio Macchia – The poset of proper divisibility

Mateusz Michalek – The poset of normal polytopes

Satoshi Murai – Average of polyhedral Morse inequalities and graded Betti numbers

Uwe Nagel – Hilbert series up to symmetry

Sara Saeedi Madani – Binomial edge ideals and determinantal facet ideals

Matteo Varbaro – When does depth stabilize early on?

Volkmar Welker – The ideal of orthogonal representations of a graph

#### Abstracts

##### Alessio Caminata – Symmetric signature

The F-signature is a measure for singularities in positive characteristic, which is based on the asymptotic splitting behaviour of the ring viewed as a module over itself via powers of the Frobenius homomorphism. In an attempt to develop a characteristic-free notion, we introduce the symmetric signature of a local ring, which is defined by looking at the asymptotic splitting behavior of the reflexive symmetric powers of the top-dimensional syzygy module of the residue field. As a first test case we look at two-dimensional quotient singularities k\llbracket x,y\rrbracket^G, where G is a finite small subgroup of \mathrm{Gl}(2,k). Using the Auslander correspondence and representation theory we prove that if G\subseteq\mathrm{Sl}(2,k) or G is cyclic, then the symmetric signature is 1/|G|, which coincides with the F-signature. This is based on joint works with H. Brenner, and with L. Katthän.

##### Alexandru Constantinescu – Linear syzygies and regularity

We show that for every positive integer r there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to r. For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for d > 4 every triangulation of a d-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is O(log(log(n)), where n is the number of variables.

##### Alessio D’Alì – Deviations of graded algebras – slides

Let R be the quotient of a polynomial ring over a field k by a homogeneous ideal containing no linear forms. The deviations of R are numerical values that can be obtained by analysing the Poincaré series of k as an R-module; on the other hand, they encode interesting information about a differential graded algebra structure on the minimal free resolution of k as an R-module, i.e. the so-called acyclic closure of k.

In this talk we will investigate the behaviour of deviations under Gröbner and lex-segment deformation and study their asymptotics for Golod rings and some special edge rings. Moreover, we will recall the connection with Koszul homology and show an application in the case of edge rings of cycles.

This is joint work with A. Boocher, E. Grifo, J. Montaño, and A. Sammartano. The project originated during the 2014 edition of the PRAGMATIC school.

##### Emanuela De Negri – A Gorenstein simplicial complex for symmmetric minors

Let X = (x_{ij}) be the n\times n generic symmetric matrix and let S=K [x_{ij} ~|~ 1 \leq i \leq j \leq n] be the associated polynomial ring over a field $K$. Denote by I_t the ideal generated by the t-minors of X. The ring S/I_t is a Cohen-Macaulay normal domain and it is Gorenstein if and only if n-t is even.

It is well known that the t-minors of X are a Gröbner bases with respect to the lexicographic order with x_{11}>x_{12}>\dots>x_{1n}>x_{22}>\dots>x_{nn}. The corresponding initial ideal is square-free and Cohen-Macaulay, but it is not Gorenstein.

The question when Gorenstein ideals have initial ideals which are square-free and Gorenstein have been answered affirmatively in several instances, for example for ideals of minors of generic matrices and for ideals of Pfaffians of skew symmetric matrices.

So far for the ideal I_t (with n-t even) only the case t=2 has been treated. In this talk the case t=n-2 is considered. We prove that the (n-2)-minors form a Gröbner basis of I_{n-2} with respect to a suitable reverse lexicographic order and that the corresponding initial ideal in(I_{n-2}) is square-free and Gorenstein. It also turns out that the Betti number of I_{n-2} and in(I_{n-2}) actually coincide.

This is a joint work with A.Conca and V.Welker.

##### Gunnar Fløystad – Letterplace and co-letterplace ideals of posets

We study two types of monomial ideals associated to a poset P, the n‘th letterplace ideals L(n,P) and the n‘th co-letterplace ideals L(P,n). These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals, or subideals of these, by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, d-partite d-uniform ideals, Ferrers ideals, edge ideals of cointerval d-hypergraphs, and uniform face ideals.

##### Takayuki Hibi – Gorenstein Fano polytopes and quadratic Gröbner bases

Certain Fano polytopes arising from order polytopes and chain polytopes of finite partially ordered sets will be discussed. Showing that these Fano polytopes possess reverse lexicographic quadratic Gröbner bases guarantees that they are Gorenstein.

This is a joint work with the coauthors of arXiv:1409.4386 (J. Algebra, to appear), arXiv:1410.4786, arXiv:1505.04289 and arXiv:1506.00802.

##### June Huh – Hodge theory for combinatorial geometries – slides

A conjecture of Read predicts that the coefficients of the chromatic polynomial of any graph form a log-concave sequence. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. All known proofs use Hodge theory for projective varieties, and the more general conjecture of Rota for possibly “nonrealizable” configurations is still open. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This cohomology theory gives strong restrictions on numerical invariants of tropical varieties, and in particular those conjectured by Rota. Joint work with Karim Adiprasito and Eric Katz.

##### Martina Juhnke-Kubitzke – Gap vectors of real projective varieties – slides

Given a totally real, non-degenerate projective variety X\subseteq \mathbb{P}^m and a generic set of points \Gamma\subseteq X(\mathbb{R}), we consider the cone P of nonnegative quadratic forms on X and the cone \Sigma of sums of squares of linear forms. We examine the dimensions of the faces P(\Gamma) and \Sigma(\Gamma) consisting of forms in P and \Sigma which vanish on \Gamma. As the cardinality of the set \Gamma varies in 1,2,\ldots, \mathtt{codim}(X), the difference between the dimensions of P(\Gamma) and \Sigma(\Gamma) defines a numerical invariant of X, the so-called gap vector of X. We study fundamental properties of this vector and provide a formula relating the components of the gap vector of X and the quadratic deficiencies of X and its generic projections. Using this formula, we prove that gap vectors are weakly increasing, obtain upper bounds for their rate of growth and show that these upper bounds are eventually achieved for all varieties. Moreover, we give a characterization of the varieties with the simplest gap vectors. This is joint work with Greg Blekherman, Sadik Iliman and Mauricio Velasco.

##### Thomas Kahle – Plethysm and lattice point counting – slides

The plethysm problem in representation theory of GL(n) is to compute the irreducible decomposition of a composition of Schur functors. We give a gentle introduction to the problem and show that the answer can be computed by lattice point counting techniques. This allows to study the asymptotics of plethysm and to derive explicit quasi-polynomial formulas for plethysm. We show the (somewhat surprising) fact that even along individual rays, plethysm counting functions are not necessarily Ehrhart functions. This poses a major problem in the search for combinatorial explanations for plethysm.

This is joint work with Mateusz Michalek.

##### Lukas Katthän – The Golod property of Stanley-Reisner rings

A graded (or local) ring is called Golod if the Betti numbers of the residue field grow as fast as possible. This is equivalent to the vanishing of all Massey products on the Koszul homology. In this talk, I will discuss the latter condition for Stanley-Reisner rings. For those rings, one has a geometric interpretation of the product on Koszul homology. It can be used to construct rings where the Golod property depends on the underlying field. Moreover, I will discuss higher Massey products and questions related to their vanishing.

##### Michal Lason – On the toric ideal of a matroid and related combinatorial problems – slides

When an ideal is defined only by combinatorial means, one expects to have a combinatorial description of its set of generators. An attempt to achieve this description often leads to surprisingly deep combinatorial questions.

White’s conjecture is an example. It asserts that the toric ideal associated to a matroid is generated by quadratic binomials corresponding to symmetric exchanges. In the combinatorial language it means that if two multisets of bases of a matroid have equal union (as a multiset), then one can pass between them by a sequence of symmetric exchanges.

White’s conjecture resisted numerous attempts since its formulation in 1980. We will discuss its relations with other open problems concerning matroids.

##### Dinh Van Le – On h-vectors of broken circuit complexes – slides

Broken circuit complexes are an essential tool in the study of various important properties of matroids and hyperplane arrangements. For instance, the characteristic polynomial of a matroid, a generalization of the chromatic polynomial of a graph, has coefficients encoded by the f-vector of the broken circuit complex of the matroid. Moreover, the broken circuit complex associated to a hyperplane arrangement can be used to construct bases for two important algebras of the arrangement, namely, the Orlik-Solomon algebra and the Orlik-Terao algebra. In this talk, I give an application of the h-vector of the broken circuit complex in characterizing the Gorenstein and complete intersection properties of the Orlik-Terao algebra. I also provide some results on h-vector of broken circuit complexes of series-parallel networks.

##### Antonio Macchia – The poset of proper divisibility

We define the order relation given by the proper divisibility between monomials, inspired by the definition of the Buchberger graph of a monomial ideal. From this order relation we obtain a new class of posets. Surprisingly, the order complexes of these posets are homologically non-trivial. We prove that these posets are dual CL-shellable, we completely describe their homology (with \mathbb{Z} coefficients) and we compute their Euler characteristic using generating functions. Moreover this relation gives the first example of a dual CL-shellable poset that is not CL-shellable. This is a joint work with D. Bolognini, E. Ventura and V. Welker.

##### Mateusz Michalek – The poset of normal polytopes – slides

Normal polytopes are combinatorial objects corresponding to embedded projectively normal projective toric varieties. Various aspects of these polytopes have been studied with tools and inspirations coming from algebra, geometry and combinatorics. In my talk I would like to present the poset structure on the family of normal polytopes, that turns out to have very interesting geometry. I will focus on recent results from a joint work with Winfried Bruns and Joseph Gubeladze providing maximal normal polytopes. I will also present several challenging open problems.

##### Satoshi Murai – Average of polyhedral Morse inequalities and graded Betti numbers – slides

In the field of combinatorial topology, Wolfgang Kuehnel developed a version of the Morse theory for simplicial complexes, which is sometimes called a polyhedral Morse theory. Recently, Bhaskar Bagchi and Basudeb Datta found that a certain average of Morse inequalities in this polyhedral Morse theory has interesting applications to the combinatorial study of triangulations of topological manifolds. This average of Morse inequalities can be written in terms of graded Betti numbers of Stanley-Reisner rings by the Hochster’s formula. In this talk, I will explain how graded betti numbers and this average of polyhedral Morse inequalities are applied to study combinatorial properties of triangulations of manifolds.

##### Uwe Nagel – Hilbert series up to symmetry

Hillar and Sullivant developed a framework for studying ideals in a polynomial ring S in countably many variables that are invariant under the action of the symmetric group or, more generally, the action of a monoid. For the monoid \mathrm{Inc} of increasing functions \mathbb{N} \to \mathbb{N}, they showed that each \mathrm{Inc}-invariant ideal in S is generated by finitely many \mathrm{Inc}-orbits. Thus, such an ideal gives rise to a chain of ideals in finitely many variables that determines the original ideal. In order to study properties of an \mathrm{Inc}-invariant ideal we associate to it a bigraded Hilbert series that is derived from a chain of ideals in finitely many variables. We show that this Hilbert series is rational.

This is based on joint work with Tim Römer.

##### Sara Saeedi Madani – Binomial edge ideals and determinantal facet ideals – slides

The study of ideals of all maximal minors of an (m\times n)-matrix X of indeterminates is a classical subject of research in Commutative Algebra. There are also some generalizations of this type of ideals. Among them is the ideal generated by a set of maximal minors of X which is called the determinantal facet ideal (of a simplicial complex). In the case m=2, this ideal is a binomial ideal called binomial edge ideal (of a graph). In this talk, we give an overview of some algebraic properties and invariants of such ideals with focus on some recent developments and open questions. Special attention is given to their minimal graded free resolution. (This talk is based on some joint works with Jürgen Herzog and Dariush Kiani).

##### Matteo Varbaro – When does depth stabilize early on? – slides

Given a homogeneous ideal I of a polynomial ring S, it is interesting to study the behavior of the depth-function of I, that is the numerical function that to a natural number k assigns \mathrm{depth}(S/I^k). By a classical result of Brodmann, the depth-function of any ideal is definitely constant. However, the initial behavior is hard to understand. During the talk, I will analyze the ideals with constant depth-function. The case in which I is radical and S/I is Cohen-Macaulay is well understood: in fact, by a result of Cowsik and Nori, in such a situation I has constant depth-function if and only if I is a complete intersection. Without these assumptions the situation is much wilder: in a recent work joint with Le Dinh Nam, we identified a class of ideals with constant depth-function, but such a class is not exhaustive in general. There still is a chance that the class above includes all square-free monomial ideals generated in a single degree with constant depth-function. I will especially discuss this question and the properties had by the ideals that we identified, which are interesting by themselves.

##### Volkmar Welker – The ideal of orthogonal representations of a graph (joint with Herzog/Macchia/Madani)

We study orthogonal representations of simple graphs G in \mathbb{R}^d from an algebraic perspective in case d=2. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to \mathbb{R}^d where non-adjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition.