This is a workshop on recent developments in arrangements. (Organizers: Takuro Abe (Fukuoka), Masahiko Yoshinaga (Sapporo))

Speakers: Benoit Guerville-Balle (Warsaw), Lukas Kuhne (Leipzig), Paul Mucksch (Bochum), Delphine Pol (Kaiserslautern), Shigetaro Tamura (Sapporo), Tan Nhat Tran (Sapporo), Shuhei Tsujie (Asahikawa), So Yamagata (Sapporo),

Short talks: Christopher de Vries (Bremen/Sapporo), Sakumi Sugawara (Sapporo), Isamu Kudo (Sapporo), Aya Takeda (Fukuoka)

Schedule (Japanese Time):

Feb. 15 (Mon)

• Morning Session 10:30-11:30 (Short talks)

• 13:30-14:30, Tan Nhat Tran,

• 14:45-15:45, Benoit Guerville-Balle,

• 16:00-17:00, Delphine Pol,

Feb. 16 (Tue)

• 10:15-10:45, Shigetaro Tamura,

• 11:00-12:00, So Yamagata,

• 13:30-14:30, Shuhei Tsujie,

• 14:45-15:45, Lukas Kuhne,

• 16:00-17:00, Paul Mucksch,

Titles and Abstracts:

2021/02/15 (Mon) 10:30-11:30 (Short talks)

Christopher de Vries, The Ehrhart quasi-polynomial of almost integral zonotopes.

Abstract: The function that counts lattice points in the n-th dilate of a rational polytope is a quasi-polynomial. If the polytope has vertices with integer coordinates, the function becomes a polynomial. These functions are called Ehrhart (quasi-)polynomials. In 1991 Stanley found a explicit description of the Ehrhart polynomial of integer zonotopes in terms of its generating vector set. We consider rational shifts of integer zonotopes, which Ehrhart quasi-polynomial was described by Ardila, Beck and McWhirther in 2020.

Aya Takeda, Graphs and strictly plus-one generated hyperplane arrangements.

Isamu Kudo, Connectivity of the realization spaces of line arrangements.

Abstract: The combinatorial method introduced by Nazir and Yoshinaga enables us to research the connectivity of realization spaces of line arrangements on \( P^2_C \), and the problem of the connectivity is generalized to arbitrary dimensions on \( C^n \) by Gallet and Saini. We will explain Nazir and Yoshinaga’s combinatorial process and discuss a generalization of the problem to higher dimensions.

Sakumi Sugawara, Handle Decomposition of Line Arrangements.

Abstract: It is known that the cell decomposition of complexified complement of the hyperplane arrangement defined over real numbers are described in terms of the real structure. In this talk, we give explicit handle decomposition.

2021/02/15 (Mon) 13:30-14:30

Tan Nhat Tran,

Title: Some probabilistic interpretations of G-Tutte polynomial.

Abstract: The notion of G-Tutte polynomial forms a unification of several Tutte-like polynomials in the literature, such as the classical Tutte polynomial, the Moci arithmetic Tutte polynomial and the characteristic quasi-polynomial. The G-Tutte polynomial was proved to have many common properties with its specializations, deriving from combinatorial and topological aspects of arrangement theory. It would be interesting to find more analogous properties. In this talk, via probabilistic aspect, we will present some interpretations of the G-Tutte polynomial as the expected value of counting functions, generalizing the classical results of Welsh (1996) on counting colorings in random graphs. Further details can be found in arXiv:2003.07349. If time permits we will mention an ongoing work related to Stanley-Reisner ring for arithmetic matroids.

2021/02/15 (Mon) 14:45-15:45

Benoit Guerville-Balle,

Title: The loop-linking number of line arrangements.

Abstract: In his PhD thesis, Cadegan-Schlieper constructs a linking invariant of the embedded topology of a line arrangements, call the loop-linking number. In this talk, we will present this invariant and give a geometrical interpretation of it. We also will describe its behavior under the union of arrangements. As an application, we will use it to prove that the complements of Rybnikov's arrangements are not homeomorphic.

2021/02/15 (Mon) 16:00-17:00

Delphine Pol,

Title: Subspace arrangements and freeness

Abstract: I will present in this talk a generalization of the notions of logarithmic vector fields and freeness for equidimensional subspace arrangements, which were first introduced by Michel Granger and Mathias Schulze in the case of complete intersections.

2021/02/16 (Tue) 10:15-10:45

Shigetaro Tamura,

Title: Postnikov-Stanley Linial arrangement conjecture and new formulas of the characteristic quasi-polynomials.

Abstract: Postnikov-Stanley conjectured that all roots of the characteristic polynomial of the Linial arrangement has the same real part. This conjecture was proved for many cases by Postnikov-Stanley, Athanasiadis and Yoshinaga. In this talk, we give new formulas of the characteristic quasi-polynomials and prove the conjecture in a special case by using these formulas.

For other cases, we explain how to verify the conjecture by a computational approach.

2021/02/16 (Tue) 11:00-12:00

So Yamagata,

Title: Combinatorics of discriminantal arrangements

Abstract: For a generic arrangement A^0 of n hyperplanes in C^k Manin-Schechtman introduced disciminantal arrangement B(n,k, A^0) as a generalization of braid arrangement. In particular, B(n,1,A^0) coincides with the braid arrangement. It is known that the arrangements can be classified into two main classes, very generic ones and non very generic ones. Though there is progress in understanding intersection lattice L(B(n, k, A^0)) for A^0 being very generic, it is not well known for non very generic ones. The first general result was given by Libgober-Settepanella in which they gave a full description of rank 2 intersections of L(B(n, k, A^0)). In this talk we will generalize their condition and give a sufficient condition for A^0 to be non very generic arrangement.This is joint work with Simona Settepanella.

2021/02/16 (Tue) 13:30-14:30

Shuhei Tsujie,

Title: A generalization of the chromatic symmetric functions for signed graphs

Abstract: Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. We introduce a generalization of the chromatic symmetric functions for signed graphs by using the integer points of the complements of signed-graphic arrangements and we verify validity of the definition. This is joint work with Masamichi Kuroda.

2021/02/16 (Tue) 14:45-15:45

Lukas Kuhne,

Title: Verifying Terao's freeness conjecture for small arrangements in arbitrary characteristic.

Abstract: This talk is concerned with computationally verifying Terao’s freeness conjecture which asserts that the freeness of a hyperplane arrangement over a fixed field is determined by the underlying matroid. In previous work, we have enumerated all matroids of rank 3 with up to 14 hyperplanes whose characteristic polynomial splits over the integers. In a current follow-up project, we investigate the moduli space of these matroids over the integers and determine the free locus within these moduli spaces using Fitting ideals. As the main result, we will present a computational proof of Terao’s freeness conjecture for rank 3 arrangements with up to 14 hyperplanes over arbitrary characteristic.This was previously known to hold in fields of characteristic 0.

This is joint work with Mohamed Barakat.

2021/02/16 (Tue) 16:00-17:00

Paul Muckcsh,

Title: On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements

Abstract: In my talk, I will explain the relationship between the cohomology of the sheaf on the intersection lattice of a hyperplane arrangement studied by Sergey Yuzvinsky in the early 1990s and the cohomology of the coherent sheaf on punctured affine space respectively projective space associated to the derivation module of the arrangement. The exact relationship is given by a Künneth formula connecting the cohomology groups, answering a question by Masahiko Yoshinaga. This, in turn, gives a new proof of Yuzvinsky’s freeness criterion and yields a stronger result, giving also a characterization of the projective dimension of the derivation module.