List of Speakers

• Takuro Abe (Kyushu University)
• Elisa Palezzato (Hokkaido University)
• Hiroaki Terao (Tokyo Institute of Technology)
• Michele Torielli (Hokkaido University)
• Tan Nhat Tran (Hokkaido University)
• Shuhei Tsujie (Hiroshima Kokusai Gakuin University)
• Masahiko Yoshinaga (Hokkaido University)

Titles and Abstracts

Takuro Abe (Kyushu University)

Title: Around free arrangements and Solomon-Terao algebras

Abstract: If we delete one hyperplane from a free arrangement, then that deleted one is either free or plus-one generated. The latter means that its logarithmic derivation module is generated by dimension plus one number of derivations with one relations, so its projective dimension is one. We study this ``around free arrangements'' problem further, and discuss the relation with singularities and Solomon-Terao algebras of them.

Elisa Palezzato (Hokkaido University)

Title: Modular approach on hyperplane arrangements

Abstract: The idea of applying modular techniques on hyperplane arrangements started by looking at the class of free arrangements. Specifically, we investigated the relations between freeness over a field of finite characteristic and freeness over $\mathbb{Q}$. This led us to study the combinatorics of hyperplane arrangements over arbitrary fields. In particular, we determined in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices. This is a joint work with M. Torielli.

Hiroaki Terao (Tokyo Institute of Technology)

Title: On the exponents of restrictions of Weyl arrangements

Michele Torielli (Hokkaido University)

Title: Associated primes and localization of hyperplane arrangements

Abstract: We will first recall the basic properties of free and plus-one generated arrangement. Then, we will describe the associated prime ideals of the Jacobian ideal of a free or plus-one generated arrangement. Moreover, we will describe how to prove that the localization of a plus-one generated arrangement is free or plus-one generated. We will conclude the talk with some open problems and ideas for future work. This is part of a work in progress with Elisa Palezzato.

Tan Nhat Tran (Hokkaido University)

Title: Eulerian polynomials for subarrangements of Weyl arrangements

Abstract: Let $\mathcal{A}$ be a Weyl arrangement. We introduce and study the notion of $\mathcal{A}$-Eulerian polynomial producing an Eulerian-like polynomial for any subarrangement of $\mathcal{A}$. This polynomial together with shift operators describe how the characteristic quasi-polynomial of a subarrangement can be expressed in terms of the Ehrhart quasi-polynomial of the fundamental alcove. It in turn generalizes two well-known formulas in the literature: the first one relates the characteristic polynomial of $\mathcal{A}$ to Ehrhart theory due to Athanasiadis (1996), Blass-Sagan (1998), Suter (1998) and Kamiya-Takemura-Terao (2010); the second one relates the number of lattice points in the fundamental parallelepiped to the Lam-Postnikov's Eulerian polynomial due to Yoshinaga (2018). We will also discuss some ongoing problems related to the $\mathcal{A}$-Eulerian polynomials via algebraic and topological aspects. This talk is based on a recent work in progress with Ahmed Umer Ashraf (Western) and Masahiko Yoshinaga (Hokkaido).

Shuhei Tsujie (Hiroshima Kokusai Gakuin University)

Title: Modular construction of free arrangements

Abstract: Stanley proved that a graphic arrangement is free if and only if the graph is chordal. Dirac showed that a graph is chordal if and only if the graph is obtained by ``gluing" complete graphs. In this talk, we formulate ``gluing" operation in terms of simple matroids and show that every arrangement obtained by the operation is divisionally free.

Masahiko Yoshinaga (Hokkaido University)

Title: Icosidodecahedron and Milnor fiber of arrangements

Abstract: In the first part, I will recall Papadima-Suciu's framework for understanding the monodromy eigenspace of the Milnor fiber cohomology in terms of Aomoto complex with finite field coefficients. Then we show that the icosidodecahedral arrangement (an arrangement of 16 planes associated with the icosidodecahedron) provides a counterexample to a part of Papadima-Suciu's conjecture. The icosidodecahedral arrangement also provide the first example whose 1st homology of the Milnor fiber has a torsion. This talk is based on arXiv:1902.06256.

Organizers

• Daisuke Suyama (Wakkanai Hokusei Gakuen University)
• Shuhei Tsujie (Hiroshima Kokusai Gakuin University)