# 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:

- the existence of homogeneous artinian ideals $I \subset k[x_0,\cdots,x_n]$ which fail the Weak Lefschetz Property;
- the existence of smooth projective varieties $X \subset \mathbb{P}^N$ satisfying at least one Laplace equation of order $s\ge 2$.

**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.

**SHIMADA, Ichiro**

*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.

# Participants

# 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).# Register

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