Workshop on Hyperplane Arrangements and Singularity Theory

22 (Tuesday) March - 25 (Friday) March 2016

Place: Department of Mathematics, Hokkaido University, Bldg no.4, 4-501 lecture room

Tentative timetable:

TUESDAY 22

10:30-11:00 Registration

11:00-12:00 Nakashima

12:00-13:30 Lunch break

13:30-14:30 Feher

14:30-15:00 Coffee break

15:00-16:00 Liao

WEDNESDAY 23

10:00-11:00 Sano

11:00-11:30 Coffee break

11:30-12:30 Kozlov

12:30-14:00 Lunch break

14:00-15:00 Tokunaga

15:00-15:30 Coffee break

15:30-16:30 Bannai

Social dinner

THURSDAY 24

9:30-10:30 Guerville-Balle

10:30-11:00 Coffee break

11:00-12:00 Shirane

12:00-13:00 Tsujie

13:00-14:30 Lunch break

Free afternoon

FRIDAY 25

10:00-11:00 Dimca

11:00-11:30 Coffee break

11:30-12:30 Feitchner

12:30-14:00 Lunch break

14:00-15:00 Delucchi

15:00-15:30 Coffee break

15:30-16:30 Abe

Titles and abstracts:

-Takuro Abe (Kyushu University)

Title: Some remarks on nearly free arrangements of lines in the projective plane

Abstract: The notion of nearly free curves and divisors were introduced by Dimca and Sticlaru recently to consider the cuspidal singularities of curves and divisiors. In this talk, we focus our interest on nearly free arrangements of lines in the projective plane, and investigate its properties, some of which are very special. We also prove that nearly free arrangements of lines is in fact "near'' to free arrangements of lines in terms of arrangement theory.

-Shinzo Bannai (Ibaraki National College of Technology)

Title: On the restricted tangent bundle of rational plane curves and existence of contact conics.

Abstract: In the study of plane curve arrangements, it is known that there are many interesting examples when contact curves are involved. However, it is difficult to determine if a given curve has a contact curve or not. In this talk, this problem will be discussed in the case of rational plane curves in relation with the restricted tangent bundles.

-Emanuele Delucchi (Fribourg University)

Title: Toric arrangements and group actions on semimatroids

Abstract: Recent work of De Concini, Procesi and Vergne on vector partition functions gave a fresh impulse to the study of toric arrangements from an algebraic, topological and combinatorial point of view. In this context, many new combinatorial structures have recently appeared in the literature, each tailored to one of the different facets of the subject. Yet, a comprehensive combinatorial framework is lacking. As a unifying structure, in this talk I will propose the study of group actions on semimatroids and of related polynomial invariants, recently introduced in joint work with Sonja Riedel.

-Alex Dimca (Nice University)

Title: A computational approach to Milnor fiber cohomology

Abstract: In this talk we consider the Milnor fiber $F$ associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.

-Laci Feher (Eotvos University Budapest)

Title: Holomorphic maps between spaces are maximally singular

Abstract: (joint work with András Némethi, Renyi Institute) A simpler variant of the phenomenon. Theorem: Let $f:P^n\to P^{n+l}$ be a non-linear holomorphic map between projective spaces. Then for any s such that s(s+l)<n+1 there is a point in $P^n$, where the kernel of the differential $df_x$ is at least s dimensional. Notice that s(s+l) is the expected cxdimension of the degeneracy locus of such points. We conjecture that the degeneracy locus of any contact singularity (with the condition that its expected dimension is non-negatice and the map is non-linear) is not empty. We can prove this generalisation in "almost all" cases. The proof relies on basic properties of equivariant cohomology and Thom polynomials. Some variants of this phenomenon are valid for real smooth maps between complex projective spaces.

-Eva Feichtner (Bremen University)

Title: Bergman fan - a link between arrangement theory and tropical geometry

Abstract: Tropicalizations of arrangement complements turn out to be rational polyhedral fans whose link at the origin is homeomorphic to the order complex of the respective intersection lattice. On the level of matroids, the so-called Bergman fans are discrete-geometric constructions that allow to recover the matroid. Proving the latter requires an intriguing mix of discrete-geometric and tropical techniques.

-Benoit Guerville-Balle (Tokyo Gakugei University)

Title: A linking invariant of algebraic curves

Abstract: We construct a topological invariant algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (i.e. a pair of curves having same combinatorics, yet different topology); and it distinguishes also a pair of Zariski pair of line arrangements. Furthermore, this last example provides the first example of arithmetic Zariski pair with non-isomorphic fundamental group.

-Dmitry Kozlov (Bremen University)

Title: Symmetry breaking labelings in standard chromatic subdivisions.

Abstract: In this talk we study combinatorics of standard chromatic subdivisions of simplices. Out main focus is to investigate the existence of symmetry breaking labelings on the vertices of an iterated standard chromatic subdivision of an n-simplex. This has important implications for the complexity estimates in distributed computing.

-Xia Liao (Kias)

Title: Hirzebruch class of singular curves

Abstract: The motivic Hirzebruch class theory is a characteristic class theory which unifies the Chern-Schwarz-McPherson class theory, singular Todd class theory and Thom-Milnor's L-class theory. In this talk, I will discuss a conjectured formula concerning the computation of the Hirzebruch class of a free divisor $D$ in a non singular variety $X$ by its sheaf of logarithmic differentials. While the conjectures formula is still difficult to understand, I will report some of my recent progress on the simplest case $X$ is a nonsingular surface and $D$ is a curve.

-Norihiro Nakashima (Toyota Technology Institute)

Title: Discrete Fourier transforms on some asymptotically good towers of curves

Abstract: An affine variety code is a generalisation of an algebraic geometry code. A low-complexity decoding algorithm for affine variety codes using the Berblekamp-Massey-Sakata (BMS) algorithm and the discrete Fourier transform (DFT) is known. At the same time, Garcia and Stichtenoth constructed a tower of curves which reaches the Drinfeld-Vladut bound. In this talk, we make a method to calculate DFT on curves. By using the method to calculate the DFT, the number of finite-field operation of the decoding algorithm is reduced if affine variety codes are associated to the curves defined by Garcia and Stichtenoth.

-Taro Sano (Kobe University)

Title: A bound of singularities on a Q-Fano 3-fold

Abstract: It is known that there are a finitely many families of Q-Fano 3-folds with terminal singularities. Thus there is a bound of number of singular points on them. In my talk, I will explain about a formula of number of singular points on a Q-Fano 3-fold by its topological data.

-Taketo Shirane (National Institute of Technology, Ube College)

Title: Splitting numbers and $\pi_1$-equivalent Zariski $k$-plets

Abstract: In 2003, I. Shimada constructed equisingular families of plane curves with many connected components, which cannot be distinguished by fundamental groups. We introduce splitting numbers of irreducible plane curves for Galois covers, and show that the splitting numbers are invariant under certain homeomorphisms of the complex projective plane. By splitting number, we prove that Shimada’s equisingular families provide $\pi_1$-equivalent Zariski $k$-plets.

-Hiroo Tokunaga (Tokyo Metropolitan University)

Title: Geometry of contact conics

Abstract: Let $D$ be a reduced plain curve. A smooth conic $C$ is said to be a contact conic to $D$ if the intersection number $l_x(C,D)$ is even for all $x\in C\cap D$ and all points in $C\cap D$ are smooth points of $D$. A simple contact conic means that "contact"+$l_x(C,D)=2$ for all $x\in C\cap D$. In this talk, we discuss some problems and results related contact conics such as Zariski pair (k-plet) and splitting curves in double covers.

-Shuhei Tsujie (Hokkaido University)

Title: Freeness for $\psi$-graphical arrangements.

Abstract: Richard P. Stanley introduced $\psi$-graphical arrangements as a generalization of graphical arrangements. After that, Lili Mu and Stanley characterized supersolvability for $\psi$-graphical arrangements and conjectured that super solvability and freeness are equivalent for $\psi$-graphical arrangements. In this talk, we will verify this conjecture in terms of vertex-weighted graphs over a poset. This talk is based on a joint work with Daisuke Suyama.

Organizers: Toru Ohmoto (Hokkaido University), Michele Torielli (Hokkaido University), Masahiko Yoshinaga (Hokkaido University).

Contact: Michele Torielli (torielli [at] math.sci.hokudai.ac.jp)