20 (Wednesday) February - 22 (Friday) February 2019
Place: Department of Mathematics, Hokkaido University, Bldg no.4, 4-501 lecture room.
List of Speakers:
- Takuro Abe (Kyushu University)
- Weili Guo (Hokkaido University)
- Takahiro Nagaoka (Kyoto University)
- Norihiro Nakashima (Nagoya Institute of Technology)
- Elisa Palezzato (Hokkaido University)
- Hiroo Tokunaga (Tokyo Metropolitan University)
- Michele Torielli (Hokkaido University)
- Tan Nhat Tran (Hokkaido University)
- Shuhei Tsujie (Hiroshima Kokusai Gakuin University)
- So Yamagata (Hokkaido University)
Tentative timetable:
WEDNESDAY 20
10:00-11:00 Tokunaga
11:00-11:30 Coffee break
11:30-12:30 Nakashima
12:30-13:30 Lunch break
13:30-14:30 Palezzato
14:30-15:00 Coffee break
15:00-16:00 Nagaoka
THURSDAY 21
9:30-10:30 Abe
10:30-11:00 Coffee break
11:00-12:00 Torielli
12:00-13:00 Guo
13:00-14:30 Lunch break
Free afternoon
FRIDAY 22
10:00-11:00 Tran
11:00-11:30 Coffee break
11:30-12:30 Tsujie
12:30-14:00 Lunch break
14:00-15:00 Yamagata
Titles and abstracts:
- Takuro Abe (Kyushu University)
Title: Combinatorics of the addition-deletion theorems for arrangements
Abstract: The most useful methods to study free arrangements is Terao's addition-deletion theorems. Based on them the inductively free arrangements has been constructed in which Terao's conjecture is true. In this talk we show that Terao's addition-deletion theorems are combinatorial, i.e., if you are given one free arrangement among the triple, then whether all of them are free or not depends only on the intersection lattice. By this combinatorialization, we can construct new classes of free arrangements called the divisionally and additionally free arrangements in which Terao's conjecture is true. Also, we will discuss the local freeness, Chern and characteristic polynomials and their multiple roots.
- Weili Guo (Hokkaido University)
Title: On the Falk Invariant of Shi-arrangement
Abstract: It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of Shi, Linial and semiorder arrangements.
- Takahiro Nagaoka (Kyoto University)
Title: Hypertoric varieties and hyperplane arrangements
Abstract: Hypertoric varieties are algebraic varieties, defined as an analogue of topic varieties. As the geometric properties of (projective) topic varieties can be studied by the associated polytopes, hypertoric varieties can be studied by its associated hyperplane arrangements. In this talk, I will discuss (Poisson) deformation of hypertoric varieties. In applications, I will classify the singularities of affine hypertoric varieties and I will count the number of non-isomorphic crepant resolutions of affine hypertoric varieties in terms of hyperplane arrangements.
- Norihiro Nakashima (Nagoya Institute of Technology)
Title: High order freeness for 3-arrangements
Abstract: The m-free arrangement is a generalisation of the free arrangement where m is a nonnegative integer. Holm asked whether all arrangements are m-free for m large enough and he proved that the question is true for 2-arrangements. In a recent work by Abe and the speaker, it is shown that the question is not true when the dimension of the vector space is grater than 3. In this talk we will discuss the question for 3-arrangements.
- Elisa Palezzato (Hokkaido University)
Title: On the freeness of hyperplane arrangements over arbitrary fields.
Abstract: In this talk I will recall the basic properties of hyperplane arrangements and their freeness. Moreover, I will describe the relations between freeness over the rationals and over fields of finite characteristic. This is a joint work with Michele Torielli.
- Hiroo Tokunaga (Tokyo Metropolitan University)
Title: A remark on certain cubic-line arrangements and elliptic surfaces.
Abstract: Geometry and arithmetic of cubic curves and elliptic surfaces have been used to construct topologically interesting plane curves, i.e., possible candidates for Zariski pairs. In this talk, we explain another such example of plane curves B_1 and B_2 as follows:
1) Both B_1 and B_2 consist of a nodal cubic and 4 lines.
2) B_1 and B_2 have the same combinatorics.
3) Both fundamental groups \pi_1(P^2\setminus B_i,*) (i=1, 2) are non-abelian.
4) (B_1, B_2) is a Zariski pair.
- Michele Torielli (Hokkaido University)
Title: On the associated primes of hyperplane arrangements
Abstract: In the first part of this talk, we will recall the basic properties of free and plus-one generated arrangement. In the second one, we will describe the associated prime ideals of the Jacobian ideal of a free or plus-one generated arrangement. 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 and Masahiko Yoshinaga.
- Tan Nhat Tran (Hokkaido University)
Title: A combinatorial description of the exponents of $A_1^2$ restrictions of Weyl arrangements
Abstract: Let A be a Weyl arrangement in an l-dimensional Euclidean space. Using a case-by-case argument, Orlik-Terao (1993) proved that any restriction of A is free. Prior to this, Orlik-Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. However, describing theoretically their exponents is still a difficult task. A classical result, due to Orlik-Solomon-Terao (1986), asserts that the exponents of any A_1 restriction i.e., the restriction of A to a hyperplane, are given by {m_1,... , m_{l-1}}, where exp(A)={m_1,... , m_l} with m_1 \le ...\le m_l. As a next step after Orlik-Solomon-Terao towards investigations on the exponents of restrictions, we are especially interested in the A_1^2 restrictions i.e., the restrictions of A to subspaces X of the type A_1^2. In this talk, we will present a description of the exponents of such restrictions in terms of the classical notion of related roots by Kostant (1955). This is a joint work with Takuro Abe and Hiroaki Terao.
- Shuhei Tsujie (Hiroshima Kokusai Gakuin University)
Title: Partitions and intersections of Shi and Catalan arrangements
Abstract: It is well known that the number of intersection of Braid arrangements are the Bell numbers. In this talk, we will discuss the number of intersections of Shi and Catalan arrangements and see that they coincide with well-known sequences. First we will review the theory of gain graphs and give a presentation of the posets of flats of the semimatroid corresponding to a gain graph in terms of a kind of set partitions and apply it to Shi and Catalan arrangements. Moreover, when the gain group is $\{\pm 1\}$, we will relate partitions with a basis for the algebra of "symmetric functions of type B". This is a joint work with Norihiro Nakashima.
- So Yamagata (Hokkaido University)
Title: Pappus’ theorem in Grassmannian
Abstract: The discriminantal arrangement was defined by Manin-Schechtman in 1989 as a generalization of braid arrangement. Its combinatorics (i.e., intersection poset) is complicated and only partial description is known. In my talk, I will explain the connection between combinatorics of discriminantal arrangement in codimension 2 and quadrics in Grassmannian.
Organizers: Takahiro Hasebe (Hokkaido University), Michele Torielli (Hokkaido University), Masahiko Yoshinaga (Hokkaido University).
Contact: Michele Torielli (torielli [at] math.sci.hokudai.ac.jp)
This program is supported by: the "Budget Support for the Leader Developing Program".