This project addresses a fundamental question on the relation between combinatorics and geometry. Mathematicians can prove theorems in combinatorics using algebraic geometry. Vice versa, we can do algebraic geometry also in absence of the algebraic variety by using combinatorics. These leitmotifs ware recently discovered by the field medalist June Huh in a linear setting.

The question is: does Huh’s philosophy depend on the linear setting? Does it apply in much broader generality? The project will answer this question by studying the toric case. This case is the middle point between the linear setting and the generic case.

The project is multidisciplinary sitting between combinatorics, geometry, algebra, and topology. PI’s knowledge in the field of arrangements and matroids is an almost unique and necessary mix to carry out this research.

The aim is to show that almost all results about matroids – obtained in the last five years using algebraic geometry – can be extended to arithmetic matroids. The research will also increase the knowledge of toric arrangements and make them a new toy example and a new local model for all geometric theories. Concretely, the research group will invent a machinery that for almost any poset produces an algebra, theorems for that algebra, and inequalities between the numerical invariants of the given poset.

Methods used in this research will be various, including sheaves on posets, Chern theory, toric and tropical geometry. The last part of the project consists in applications of the machinery to combinatorics and algebraic geometry; in particular to polytope theory, resolution of singularities and moduli spaces.

The main idea is to define a class of posets that behaves like the poset of flats of a toric arrangement; from that poset construct a Chow ring of the poset, prove the Hodge package, the relative Lefschetz decomposition, and describe the NEF cone. Then, the research group will define and study important classes in the Chow ring of the poset such as CSM classes and Chern classes of a tautological bundle.

Finally, the intersection numbers of these classes will be related to the combinatorics of the poset. The aforementioned theorems, inspired by the algebraic geometry, will provide new inequalities for such numerical invariants.

MOTIVATION

Hyperplane arrangements is an interdisciplinary topic that correlate with combinatorics, algebra, topology, and algebraic geometry. It was studied intensively but separately from these four points of view. In the last years, new ideas appear in this research field: the field medalist June Huh shows a new way to mix these four areas. Inspired by algebraic invariants of the topology of certain algebraic varieties, he constructed algebraic objects from combinatorial ones and proved theorems about these algebraic objects. As consequence of these theorem, he solved long-standing conjectures on the starting combinatorial object.

More precisely, consider the wonderful compactification of a hyperplane arrangement complement and its cohomology and Chow rings. It satisfies three famous theorems: Poincarè duality, Hard Lefschetz theorem and Hodge Riemann relations. To every hyperplane arrangement is associated a combinatorial object called matroid. But there exist matroids that do not arise as the combinatorics of any hyperplane arrangement. It has been proven that (asyntotically) the 0% of matroids came from geometric objects (hyperplane arrangements). The main idea is to define a ring for every matroid, called the Chow ring of the matroid, and to prove the three theorems. There are several proofs of such theorems: the first one is algebraic, but later were proved using combinatorics or toric geometry. The Hodge Riemann bilinear relations imply the unimodality of the Whitney numbers of a matroid. This unimodal behavior was conjecture by Rota et al. in the sixties.

Nowadays, there is prolific research in this area. Mainly focused on defining other rings associated to matroids (e.g., augmented Chow ring, conormal Chow ring, Chow ring of the tautological bundle, …), proving other theorems (e.g., Hirzebruch-Riemann-Roch theorem) and deducing other inequality (e.g., top-heavy conjecture, strong Mason’s conjecture, nonnegativity of matroid Kazhdan–Lusztig polynomials, log-concavity of h- and f-vectors of independent complex and broken circuit complex).

This project wants to push the research in a different direction. Let us ask: why we consider hyperplane arrangements and matroids? The answer to this question is complicate and it requires a very deep knowledge of the topic. However, two motivations are very clear: firstly, we have an almost complete knowledge of hyperplane arrangements. Secondly, in this restricted setting of matroids we can obtain and hope for stronger and more elegant inequalities.

Other possible settings are:

·  subspace arrangements/polymatroids,

·  toric arrangements/arithmetic matroids, and

·  arrangements of nice subvarieties/certain posets

Polymatroids are the combinatorial object associated to arrangements of subspaces of arbitrary codimension in a given vector space. Recently, it has been shown that the principal constructions and theorems hold also in this setting, but the inequalities fail due to the higher codimension. The main and common feature of hyperplane arrangement and subspace arrangement combinatorics is that intersection of two linear subspace is connected. The goal of this project is to show that linearity and connectedness are not necessary for the aforementioned constructions.

The ultimate goal would be to explore the setting of arrangements of subvarieties in a given algebraic variety such that those subvarieties intersect locally as subspace arrangements. In other words, we exclude tangent behavior and singularities of the subvarieties. But it is unrealistic to face such a general case without middle steps.

The goal of this project that will last three years is to extend the theory from the linear to the toric setting.

ACTIVITIES AND OBJECTIVES

The research activities are divided in three Milestones. The first one is the preparation: we need to push further the research on toric arrangements that are the realizable case. The second step consists in extend constructions and theorems to the non-realizable case. Finally, the last milestone consists in exploring numerical invariants and inequalities between them, as well as applications to other subjects.

Milestone A) Toric arrangements.

Firstly, consider the geometric model, i.e., wonderful model of toric arrangements. It depends on two choices: a building set and a good toric variety. The aim is to understand its cohomology ring in general. It was already described in a particular case (well-connected building set), but doing such assumption is equivalent to require that the intersection of some divisors in the toric wonderful model is always connected (or empty). This assumption is necessary in the De Concini and Gaiffi description of the cohomology ring, but our goal is to overcome the connectedness setting.

Objective A.1: Describe the cohomology of toric wonderful model for arbitrary building set.

Once described the cohomology/Chow ring of the toric wonderful model by generators and relations, we investigate the Hard Lefschetz theorem and the Hodge Riemann relations. They hold for any class in the ample cone, but in Milestone B we need to prove such theorems. Morally, we will need to prove that the “ample cone of an arithmetic matroid” is nonempty. The first step is to provide a description of it.

Objective A.2: Give a combinatorial description of the ample and NEF cones of the toric wonderful model.

In algebraic geometry, there exists varieties with very bad properties. For example, the ample cone cannot be described with a finite number of inequalities. Or the ample cone maybe empty and the NEF cone can be nonempty. We hope that toric wonderful models have a polyhedral ample cone. If this were not the case, for our porpoise it is sufficient to find a polyhedral cone inside the ample cone (that is certainly nonempty).

We want to show that other machineries developed for matroids can be generalized to the toric case. Since there is a proliferation of similar constructions, we select two of them: tautological classes and relative Lefschetz decomposition. These two are very important in the theory of matroids and have important consequences. Tautological classes of matroids can be specialized to the Chern-Schwartz-MacPherson classes and provide a Tutte polynomial in four variables whose coefficients form a log-concave sequence. Relative Lefschetz decomposition provides the intersection cohomology module of matroids and the singular Hodge theory on such modules. It was used to prove the top-heavy conjecture and the nonnegativity of Kazhdan–Lusztig polynomials.

Objective A.3: Describes the CSM classes and tautological classes of toric arrangements.

In principle the CSM classes of toric wonderful model can be computed directly and hence this objective is feasible. On the other hand, the risk of this point is intrinsic in the definition of the tautological bundle. Does it can be defined also for toric arrangements? If so, we can compute the Chern classes of the tautological bundle and specialized them to the CSM classes.

Objective A.4: Understand the natural maps between toric wonderful models. In particular, at cohomological level, provide a relative Lefschetz decomposition.

This objective has no intrinsic obstacles; however, it is very likely that will be difficult to obtain. Indeed, the geometry of the ambient space (an iteration of blowups of a toric variety along non-toric subvarieties) will cause a lot of technical issue, which do not appear in the linear case.

Milestone B) Non realizable case.

The second step will be to extend the achievements in Milestone A to the non-realizable setting. There are four models that codify the combinatorics of toric arrangement extending it to non-realizable case. In the previous studies on toric arrangements, the algebraic invariants – such as cohomology rings of the complement or the chow ring of toric wonderful model – depend on the poset of layers, i.e., the connected components of intersection of some hypertori from our arrangement. The four model are:

• arithmetic matroids

• matroids over Z

• Z-semimatroids

• matroid schemes

The first two models do not have enough power to describe the poset of layers. The last two codify the poset of layers, but those theories lack of duality and arithmetic properties.

Objective B.1: Decide what is the right combinatorial object to codify the combinatorics of toric arrangements.

It is also possible that the sought combinatorial object is not in the above list, in this case the research group will define it and study its properties. In the worst scenario, a combinatorial object with all the desired properties do not exists.

Then, we will focus on the dependence from the choice of a good toric variety. Since the Chow ring depends on this choice, the same is supposed to be for all classes (Chern, CSM, tautological) that lie in the Chow ring. Therefore, we consider the limit of all this models (as previously done by De Concini and Gaiffi) and extend the results in Milestone A to the infinite model.

Objective B.2: Study the dependency of the wonderful model from the choice of a good toric variety and extend the results in Milestone A to the infinite model.

Suppose that we are not in the worst scenario of Objective B.1; then we can put together the two previous objectives to prove the entire theory in the combinatorial case:

Objective B.3: Extend the constructions in Milestone A to the non-realizable case.

Until now all the proofs can be based on the geometry of varieties, the main difficult in Objective B.3 is proving it without using geometry.

Milestone C) Applications.

The last part of the project is focused on applications and numerical inequalities. We will explore gamma-nonnegativity, logconcavity, ultra-logconcavity, and unimodality of numerical sequences associated to our combinatorial objects.

Objective C.1, application to combinatorics: Study log-concavity of polynomial invariants associated to the combinatorial structure determined in Objective B1.

The sequences considered will include – but not limited to – arithmetic Tutte polynomial, h-vectors and f-vectors of independence complex and broken circuit complex, and Kazhdan–Lusztig polynomials for the toric case.

As application, we try to make the resolution of embedded singularities explicit (under some hypothesis) with particular attention to the cohomology ring of the complement.

Objective C.2, application to resolution of singularities: Provide an effective method to present the cohomology of a nice open set in smooth algebraic varieties.

The assumption will be that the singularities locally look like a subspace arrangement. The main issue is to put together all these local behaviors and provides a global description. The easiest nontrivial example of such problem is a toric arrangement! If our new theory would be robust enough, then Objective C.2 will follow with a small effort.

Finally, our research will have an impact on the moduli space: we will be able to enumerate the universal compactified Jacobians.

Objective C.3, application to moduli space: count the universal compactified Jacobians.

By their description, these universal compactified Jacobians depends on the choice of a parameter. This parameter lies in the complement of an affine periodic hyperplane arrangement. Count them up to translations by the integer lattice, is a problem naturally states in terms of toric arrangements.