Hyperplane Arrangements, Wonderful Compactifications, and Tropicalization (14frg193)

Arriving in Banff, Alberta Sunday, April 20 and departing Sunday April 27, 2014

Organizers

(University of Western Ontario)

(University of Michigan)

(Northeastern University)

Objectives

Rota's conjecture and the missing axiom of matroid theory

Various classical inequalities conjectured to hold for coefficients of chromatic polynomials of graphs were recently established by Huh [Hu] and Huh--Katz [HK]. When viewed from the framework of tropical geometry, the proof in [Hu,HK] suggests a promising direction to study a more general conjecture of Rota on coefficients of characteristic polynomials of matroids.

It turns out that the algebro-geometric proof does not work in Rota's more general setting for only one reason: not every matroid is realizable as an arrangement of hyperplanes in a vector space. Mathematicians since Hilbert, who found a finite projective plane which is not coordinatizable over any field, have been interested in this tension between the axioms of combinatorial geometry and algebraic geometry. After numerous unsuccessful quests for the ``missing axiom'' which guarantees realizability, logicians found that one cannot add finitely many new axioms to matroid theory to resolve the tension [MNW,Va]. On the other hand, computer experiments revealed that numerical invariants of small matroids behave as if they were realizable, confirming Rota's conjecture in particular for all matroids within the range of our computational capabilities.

Arguments of [HK] reveal that there is a weaker `tropical' realizability condition on matroids which imposes the same numerical restrictions on their characteristic polynomials as the classical realizability condition. We plan to explore the intriguing possibility that all matroids satisfy the weaker condition on realizability. If true, it will not only prove old conjectures on coefficients of characteristic polynomials of matroids, but also explain the subtle discrepancy between combinatorial and algebraic geometry.

Hyperplane arrangements and Milnor fibrations

Let $A$ be a finite collection of hyperplanes in $C^{ell}$: i.e., a complex-linear matroid realization. A key question in the field is to elucidate the relationship between the geometry of the complement and the matroid of $A$, see for instance the book-in-progress [CDFSSTY]. The paradigmatic result is the combination of two theorems, by Arnol'd--Brieskorn and Orlik--Solomon, showing that the cohomology ring of the complement is determined by the matroid. It is known that complexified real arrangements with isomorphic oriented matroids have homeomorphic complements, but that complex arrangements with isomorphic matroids may have non-homeomorphic complements.

The union of the hyperplanes in $A$ has in general a non-isolated singularity at the origin of $C^{ell}$. The topology of the Milnor fibration of the singularity has been the object of intense study in the past two years, in work of Denham, Papadima, Suciu [DS13, PS13, Su13] and many others. As a result, it is now known that the homology groups of the Milnor fiber of a hyperplane arrangement may have non-zero torsion, and that the first Betti number of the Milnor fiber is combinatorially determined, provided certain multiplicity assumptions are satisfied.

The unifying framework for the topological study of hyperplane arrangements and their Milnor fibers is provided by the characteristic and resonance varieties of their complements. The subtle interplay between the geometry of these varieties in positive versus zero characteristic accounts for much of the recent progress in this direction. We expect that the toric compactification and tropical geometric point of view will allow us to make further progress towards solving the long-standing problem of determining the Betti numbers of the Milnor fibration of an arrangement. An intermediate objective towards this aim is to construct a semistable degeneration of the tropical compactification of the Milnor fiber of an arrangement, a `wonderful model' of the Milnor fibration.

The work of Papadima and Suciu [PS13] on Milnor fibrations of arrangements reveals some unexpected differences between realizable and non-realizable matroids. For instance, the modular Aomoto--Betti numbers $\beta_p(M)$ associated to the Orlik--Solomon algebra of a matroid $M$ are stringently constrained in the realizable case, at least at the prime $p=3$. We plan to further explore the delicate boundary between realizability and non-realizability in this context, with the aim of elucidating the range of the invariants $\beta_p(A)$, for arbitrary hyperplane arrangements $A$ and primes $p$.

Wonderful and tropical compactifications


Here is more detail about recent developments related to our main toolkit. The wonderful models of De Conci-ni and Procesi [DP] are compactifications of arrangement complements, with well-behaved boundaries (normal crossings divisors) and somewhat delicate but well-understood combinatorics that describes their structure. The complement of a hyperplane arrangement is simply the intersection of a linear space with an algebraic torus. Feichtner and Sturmfels showed in [FS] that the wonderful models could be constructed by compactifying the tori inside suitable toric varieties. Such a construction is an instance of a tropical compactification, as introduced by Tevelev in [Te].

Using this point of view, the machinery of toric geometry can be brought to bear, the most comprehensive single source for which is the book [CLS]. Using the observation that the wonderful models are subvarieties of toric varieties, for example, Keel and Tevelev computed in [KT] equations for the compactified moduli space $\overline{M}_{0,n}$. It seems likely that a generalization of their work to other wonderful models could take place, by analyzing ideals in the Cox rings of the Feichter--Yuzvinsky toric varieties [FeY], and other examples of tropical compactifications that arise naturally in related settings. In this context, Block and Maclagan [BM] give an invariant ring description of these Cox rings, generalizing the Doran--Giansiracusa description for $\overline{M}_{0,n}$, as well as the associated power ideal description of the graded pieces of the Cox rings. It remains to be seen what role realizability plays here, since their description, a priori, depends on both the matroid combinatorics and a choice of linear realization.

Conclusion


In view of rapid advances in tropical geometry and recent breakthroughs proving some classical conjectural inequalities, there have been some recent informal discussions amongst the proposed participants centered around the ideas above. We feel that it would now be a strategic time to meet in a relatively small group to work intensively on advancing these projects.

[1] http://arxiv.org/abs/#1

Bibliography


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