Arrangements of plane curves and related problems

16 (Mon) -- 18 (Wed) March, 2015.
Tokyo Metropolitan University, Mnami-Osawa Campus (Room 610, 6F Bldg 8, (86K610)).

This workshop is supported by JSPS Grants-in-Aid for Scientific Research 25610007 (Challenging Exploratory Research), JSPS-MAE Sakura Program "Geometry and combinatorics of hyperplane arrangements and related topics" and Reseach Grant (Tokyo Metro University).

Invited Speakers: Schedule (Tentative)
16 (Mon) March 2015
10:00-11:00, Oka
11:15-12:15, Bailet
14:00-15:00, Libgober 1
15:15-16:15, Tokunaga

17 (Tue) March 2015
10:00-11:00, Abe
11:15-12:15, Bannai
14:00-15:00, Libgober 2
15:15-16:15, Callegaro

18 (Wed) March 2015
10:00-11:00, Settepanella
11:15-12:15, Yoshinaga




Abe
Title: Division free theorem for line arrangements and divisionally free arrangements of hyperplanes
Abstract: The most useful method to show the freeness of line arrangements in the projective space is Terao's addition-deletion theorems. First we state a combinatorial formulation of them, i.e., a line arrangement is free if its Poincare polynomial is divisible by that of its restriction. Starting from this case, we propose a new class of free arrangements, called divisionally free arrangements of hyperplanes. This class contains the calss of inductively free arrangemetns, is strictly larger than it, and Terao's conjecture holds in it.

Bailet
Title: Degeneration of Orlik-Solomon algebras and Milnor fibers of complex line arrangements.
Abstract: PDF

Bannai
Title: Splitting curves of double coverings of the projective plane.
Abstract: In this talk, we study certain double coverings of the projective plane and splitting curves, i.e., irreducible plane curves whose pre-images under the double-cover map become reducible. We introduce the notion of a "splitting type" and apply it to study the topology of curve arrangements. This is joint work with T. Shirane.

Callegaro
Title: Cohomology ring for toric arrangements
Abstract: The topic of this talk is the integer cohomology ring of the complement of a real complexified toric arrangement. We will recall some basic combinatorial invariants and we will show how these can help to give a presentation of the toric analogous of the Orlik-Solomon algebra. One of the main techniques involved is the Leray spectral sequence. In the case of a non-unimodular arrangement, it is still an open problem to find a suitable combinatorial object that can determine the integer cohomology ring. This is joint work with E. Delucchi.

Libgober (2 talks)
Title: Abelian varieties and plane algebraic curves I.
Abstract: The goal of these two talks is to discuss the topological and Hodge theoretical invariants associated with the complements to arrangements of plane algebraic curves. In the first lecture I will focus on plane curves with nodes, cusps or ADE as the only singularities and discuss the problems about Albanese varieties of cyclic covers of plane ramified over singular curves. One of the motivating problems is to find characterization of the polynomials p(t) such that there exist plane algebraic curve with Alexander polynomial p(t). Part of the discussion is a description of Mordell-Weil groups associated with plane singular curves. Several examples are dealing with arrangements of lines.

Title: Abelian Varieties and plane algebraic curves II
Abstract: I will discuss abelian varieties associated with plane curves singularities and their role in description of Albanese varieties of abelian covers associated with plane curves and descriptions of pencils composed of components of arrangements of plane curves. In many cases one is lead to abelian varieties of CM type appearing prominently in number theory. No familiarity with abelian varieties will be assumed in either of these talks. Part of material of these lecture is based join papers with J.I.Cogolludo-Agustin and E.Artal-Bartolo.

Oka
Title: On the fundamental groups of non-generic $\mathbb{R}$-join-type curves.
Abstract: PDF

Settepanela
The nbc minimal complex of supersolvable arrangements
Abstract: PDF

Tokunaga
Title: Topology of plane curves of low degree via Galois covers and rational elliptic surfaces
Abstract: We consider existence and non-existence of many Galois covers at one time in order to study the topology of plane curves. It is rather classical to make use of Galois covers to consider the topology of plane curves, but our approach is new. In this talk, we show that the approach is effective. Bisections of certain rational elliptic surface play important roles to give explicit examples.

Yoshinaga
Title: Milnor fibers of real line arrangements.
Abstract: We give an algorithm computing monodromy eigenspaces of the first cohomology of Milnor fibers of real line arrangements. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enables us to generalize a vanishing result of Libgober, give new upper-bounds and characterize the A_3-arrangement in terms of non-triviality of Milnor monodromy.
Notice (7 Jan.): The workshop will start in the morning 16th (Mon), the finall talk will be around noon 18th (Wed). We will have free discussion in the Wednesday afternoon.
Notice (14 Jan.): Room has been changed. The new place is Room 610, 6F Bldg 8.
Notice (9 Feb.): Titles and abstracts.
Organizers:
Hiro-o Tokunaga (tokunaga (at) tmu.ac.jp)
Masahiko Yoshinaga (yoshinaga (at) math.sci.hokudai.ac.jp)