Seminar
Dijon  Basel
June 67, 2016
Institut de Mathématiques de Bourgogne ~ DIJON
Program

Automorphisms of affine del Pezzo threefolds, log NoetherFano inequality and supermaximal singularity
Ivan CHELTSOV (University of Edinburgh)
Abstract
We show how to study automorphisms of affine del Pezzo threefolds using new log version of the classical NoetherFano inequality
and Pukhlikov's supermaximal singularity.

Automorphisms of complements of smooth hypersurfaces: a landscape view
Adrien DUBOULOZ (Université BourgogneFranche Comté)
Abstract
After a brief presentation of classically known results concerning the structure of the automorphism groups
of complements of smooth non degenerate hypersurfaces in projective spaces, I will mainly present general strategies
and partial results towards the understanding of the automorphism group of the complement of a smooth cubic surface
in $\mathbb{P}^3$, and more generally those of the complements of del Pezzo surfaces of degre 1 and 2 in suitable
weighted projective spaces.
(This is intended to be an introduction to Ivan Cheltsov's Talk)

On the strong complement problem in dimension 2
Mattias HEMMIG (Universität Basel)
Abstract
Given two closed irreducible curves in the affine plane that have isomorphic complements, one can ask whether those curves are isomorphic.
This question was posed by Hanspeter Kraft among his "Challenging problems in affine nspace" (1995). A stronger version asks whether the
curves are equivalent by an automorphism of the plane. In this talk, which is based on work in progress with JeanPhilippe Furter, we address this
second question, giving a negative answer by presenting a family of counterexamples. We also discuss some special cases where the answer is positive.

Covariants, derivation invariant subsets, and first integrals
Hanspeter KRAFT (Universität Basel)
Abstract
Given an ordinary differential equation $\dot x = \xi(x)$ on a manifold $M$ where $\xi$ is a smooth vector field on $M$,
a subset $Y \subset M$ is called invariant if, for every $y \in Y$, the integral curve through $y$ defined by the
flow of the ODE belongs to $Y$. Now assume that a group $G$ is acting on $M$. A basic question is to describe the subsets $Y \subset M$
which are invariant with respect to all $G$symmetric ODE's, i.e. those corresponding to $G$symmetric vector fields $\xi$.
Another important question concerns the first integrals, i.e. the solutions of $\xi f=0$ for all $G$symmetric vector fields $\xi$.
Following GrosshansScheurleWalcher, we will consider the "algebraic" situation where $M=V$ is a complex vector space,
the vector fields are polynomial, and $G \subset GL(V)$ is an algebraic group. Then one has the following
Proposition: A closed subvariety $Y \subset V$ is invariant under all $G$symmetric vector fields $\xi$ if and only if it is stable under the action
of the semigroup $End_G(V)$ of $G$equivariant endomorphisms.
In order to study the first integrals, one can construct a "generic" quotient $X /End_G(V)$ such that the rational functions on $X / End_G(V)$
correspond to the first integrals on $X$ where $X$ is a $G$stable and invariant closed subvariety $X$ of $V$. As an application, the method above allows
to give a complete description of the first integrals for the nullcone $N_d \subset V_d$ of the binary forms $V_d$ of degree $d$,
with respect to the group $SL_2$.
This is joint work with Frank D. Grosshans.

Birational maps of del Pezzo fibrations and alphainvariants of del Pezzo surfaces
Jihun PARK (Pohang University)
Abstract
Using the alphainvariants of del Pezzo surfaces, I explain why a del Pezzo fibration of degree at most 4
with nonsingular special fiber cannot be birationally transformed into another del Pezzo fibration with nonsingular special fiber.

Embeddings of horospherical homogeneous spaces into algebraic stacks
Ronan TERPEREAU (Max Planck Institute)
Abstract
The study of equivariant embeddings of tori into algebraic varieties, also known as toric varieties,
is a wellknown topic of algebraic geometry. In a recent work, Geraschenko and Satriano considered the equivariant
embeddings of tori into algebraic stacks and proved that they are always quotient stacks of toric varieties.
In this talk, I will explain the idea of their proof, give some examples, and also explain how their result might
extend to the larger class of equivariant embeddings of horospherical homogenous spaces into algebraic stacks.

Categorification of the Weyl and Heisenberg algebras (after Khovanov)
Emmanuel WAGNER (Université BourgogneFranche Comté)
Abstract
Participants
 R. BignaletCazalet (Dijon)
 J.K Canci (Basel)
 I. Cheltsov (Edinburgh)
 A. Dubouloz (Dijon)
 D. Faenzi (Dijon)
 A. Fanelli (Basel)
 J.P. Furter (Basel)
 M. Hemmig (Basel)
 H. Kraft (Basel)
 L. MoserJauslin (Dijon)
 J. Nagel (Dijon)
 J. Park (Pohang)
 P.M. Poloni (Bern)
 R. Terpereau (Max Plank)
 E. Wagner
 S. Zimmermann (Basel)
Schedule
Monday 
Tuesday 

DUBOULOZ 09:3010:00 

CHELTSOV 10:3011:30

Lunch Break 
TERPEREAU 14:0015:00 
PARK 13:3014:30 
WAGNER 15:1516:15 
HEMMIG 14:4515:45 
KRAFT 16:4517:45 
Social Picnic 18:00 .. 
All talks on the Monday and the Social Picnic will take place in René Baire Room, 4th Floor.
On Tuesday, the talks will take place in Room 318, 3rd Floor.
The time table and the abstracts are aslo available as a pdf file .
Practical Informations
Organizers
Jérémy Blanc (Basel)
Adrien Dubouloz (Dijon)
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