GEOMETRY, TOPOLOGY AND COMBINATORICS OF HYPERPLANE ARRANGEMENTS AND RELATED PROBLEMS

Schedule

 Monday Tuesday Wednesday Thursday Friday 10h Rosa Miró-Roig Ichiro Shimada Juan Viu-Sos Hiroo Tokunaga 11h Coffe Break Coffe Break Coffe Break 11h30h Miguel Marco Jean Vallès José I. Cogolludo Enrique Artal 12h30 Jacky Cresson Laurentiu Maxim Delphine Pol 15h Michel Granger Vincent Florens Ignacio Luengo Simón Isaza 16h Dan Avritzer Botong Wang Visit to Town Michael Lönne

Talks

LÖNNE, Michael
$\pi_1$ of discriminant complements of isolated quasi-homogeneous hypersurface singularities of chain type
A distinguished Dynkin diagram for the singularities of the title, $x_1^{k_1}x_2+x_2^{k_2}x_3 \cdots x_{n-1}^{k_{n-1}}x_n +x_n^{k_n}$ can be obtained by a method given by Gabrielov. It encodes the algebraic monodromy and thus information on the discriminant knot group, the fundamental group of the discriminant complement. We will first explain how the diagram of Gabrielov encodes a finitely presented group $\pi= \pi(k_1,k_2,\ldots, k_n)$. A closer look at a very degenerate plane section of the discriminant will then lead to a strategy to show that $\pi$ is the discriminant knot group. Finally we will sketch a complete proof of the claim in case $n=2$.

MAXIM, Laurentiu
Motivic infinite cyclic covers

Abstract: To an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold we associate (assuming certain finiteness conditions are satisfied) an element in the equivariant Grothendieck ring of varieties, called motivic infinite cyclic cover, which satisfies birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively. This is joint work with M. Gonzalez Villa and A. Libgober.

MIRÓ-ROIG, Rosa
The classification of smooth Togliatti system of cubics

The goal of my talk is to establish a close relationship between a priori two unrelated problems:
1. the existence of homogeneous artinian ideals $I \subset k[x_0,\cdots,x_n]$ which fail the Weak Lefschetz Property;
2. the existence of smooth projective varieties $X \subset \mathbb{P}^N$ satisfying at least one Laplace equation of order $s\ge 2$.
These are two longstanding problems which lie at the crossroads between Commutative Algebra, Algebraic Geometry, Differential Geometry and Combinatorics. In the toric case, I will classify some relevant examples and as byproduct I will provide counterexamples to Ilardi's conjecture. Finally, I will classify all smooth Togliatti system of cubics and solve a conjecture stated in my joint work with Mezzetti and Ottaviani. All I will say is based in joint work with either E. Mezzetti and G. Ottaviani or M. Michalek.

POL, Delphine
Logarithmic residues along plane curves

Reduced plane curves are always Saito free divisors, and thanks to a result of M.Granger and M.Schulze, we know that the module of logarithmic residues is the dual of the Jacobian ideal. I will give some consequences of this duality, in particular, I will explain the symmetry I have proved between the set of values of logarithmic residues and the Jacobian ideal, which is in fact a generalization of the symmetry of the semigroup of reduced reducible plane curves proved by F.Delgado de la Mata.

On the topology of projective subspaces in complex Fermat varieties

Let $X$ be the complex Fermat variety of dimension $n=2d$ and degree $m>2$. We investigate the submodule of the middle homology group of X with integer coefficients generated by the classes of standard d-dimensional subspaces contained in $X$, and give an algebraic, or rather combinatorial, criterion for the primitivity of this submodule.
This is a joint work with Alex Degtyarev.

TOKUNAGA, Hiroo
Integral sections of rational elliptic surfaces and contact conics to an irreducible $3$-nodal quartics.

Let $\varphi:S\to\mathbb{P}^1$ be a rational elliptic surface with a distinguished section $O$. A section $s$ of $\varphi$ is said to be integral if $s$ does not meet $O$. In this talk, we study integral sections of certain rational elliptic surfaces. As an application, we study geometry of contact conics to an irreducible $3$-nodal quartic. This is a joint work with K. Tumenbayar.

VALLÈS, Jean
Free divisors in a pencil of curves

A projective plane curve $D$ on a field of characteristic zero is free if its associated sheaf $\mathcal T_D$ of vector fields tangent to $D$ is a free $O_{\mathbb{P}^2}$-module. Relatively few free curves are known. I will prove that the union of all singular members of a pencil of plane projective curves with the same degree and with a smooth base locus is a free divisor, as it was conjectured by E. Artal and J.I. Cogolludo.

Organizers

The conference is organized by the Université de Pau et des Pays de l'Adour and the Instituto Universitario de Matemáticas y aplicaciones, from the Universidad de Zaragoza.

Financial supporters

We would like to thank the financial support from the following institutions and grants: GDR CNRS 2945 "Singularités et Applications", GDR CNRS 3064 "Géométrie Algébrique et Géométrie complexe", ANR GEOLMI ANR-11-BS03-0011, PHC-SAKURA 31944VE, Laboratoire de Mathématique de Pau (LMAP).

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