Mini-Workshop
Algebraic Geometry
March 23-24, 2017
Institut de Mathématiques de Bourgogne
Program
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The Picard group of the forms of the affine line and of the additive group
Raphaël ACHET (Institut Fourier, Grenoble)
Abstract
With the recent progress in the structure of linear algebraic groups over an imperfect field, it seems to be possible to study their
Picard group if the Picard groups of unipotent algebraic groups are known well enough. As every unipotent smooth connected algebraic
group is an iterated extension of forms of the additive group, this motivates the study of the Picard group of forms of the additive group.
We obtain an explicit upper bound on the torsion of the Picard group of the forms of the affine line and their regular completions.
We also obtain a sufficient condition for the Picard group of the forms of the affine line to be nontrivial and we give examples of
nontrivial forms of the affine line with trivial Picard groups.
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Birationality of quadrics
Rémi BIGNALET-CAZALET (IMB Dijon)
Abstract
To an hypersurface $X$ of $\mathbb{CP}^n$ with a defining homogeneous
polynomial $f$ in $n+1$ variables, one can associate the rational
transformation of $\mathbb{CP}^n$, called the polar of $X$, by taking the linear
system of the $n+1$ partial derivatives of $f$ and ask what can be the
topological degree of this transformation. A generalization of this
process is to take the maximal minors of the jacobian of $n$ homogeneous
polynomials in $\mathbb{CP}^n$ to obtain a transformation from $\mathbb{CP}^n$ to $\mathbb{CP}^n$. After
recalling the "hypersurface" case and explaining the generalization, we
will study what can be the topological degree of such a generalized
polar transformation focusing on the case of n homogeneous polynomials
of degree 2.
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A semistable Lefschetz (1,1)-theorem in equicharacteristic
Ambrus PAL (Imperial College, London)
Abstract
By using some elementary properties of the logarithmic de Rham-Witt complex I will explain how to prove that a rational line bundle on
the special fibre of a proper, semistable scheme over a power series ring $k[[t]]$ in characteristic $p$ lifts to the total space if and only
if its first Chern class does. This generalises a result of Morrow in the smooth case, and provides an equicharacteristic analogue of
a result of Yamashita. I will also explain a corollary concerning algebraicity of cohomology classes on varieties over global function fields.
This is joint work with Chris Lazda.
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Some geometric methods to get stability results in representation theory, application to the Heisenberg product
Maxime PELLETIER (Intitut Camille Jordan, Lyon)
Abstract
The branching coefficients, which form an important class of coefficients studied in representation theory, are by definition
the multiplicities of the irreducible representations of a connected complex reductive group $G$ inside an irreducible representation
of a similar group containing $G$. They have an interesting geometric interpretation: they can be seen as the dimensions of the spaces
of $G$-invariant sections of some $G$-linearised line bundles over some projective $G$-varieties.
This interpretation allows to demonstrate or redemonstrate some stability results on the branching coefficients, thanks to notions of geometric
invariant theory and results by Luna, and Guillemin and Sternberg. One can also obtain some explicit bounds of stabilisation in not
too difficult cases.
The end of the talk will then consist in discussing an application of these methods to the Heisenberg product,
introduced recently by Aguiar, Ferrer Santos, and Moreira. It gives rise to the Aguiar coefficients, which can be interpreted as branching
coefficients.
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Caractérisations des singularités libres
Delphine POL (LAREMA Angers)
Abstract
Dans son article fondamental, K. Saito introduit les notions de
formes et champs de vecteurs logarithmiques le long d'une hypersurface
réduite singulière. Lorsque le module des champs de vecteurs
logarithmiques est libres, on dit que l'hypersurface est libre. C'est le
cas par exemple des courbes, des diviseurs à croisements normaux ainsi
que des discriminants de singularités isolées d'intersections complètes.
Une généralisation de la notion de formes logarithmiques aux
intersections complètes réduites est développée par A.G. Aleksandrov et
A. Tsikh, puis est étendue ensuite aux espaces de Cohen-Macaulay.
L'objectif de cet exposé est d'étudier une généralisation de la notion
de liberté aux espaces de codimension plus grande.
Nous commencerons par rappeler la théorie de Saito le long des
hypersurfaces, et nous nous intéresserons ensuite aux intersections
complàtes. Nous donnerons différentes caractérisations de la liberté
pour les intersections complètes qui généralisent le cas hypersurface.
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Characterisation of affine toric varieties by their automorphism groups
Andriy REGETA (Institut Fourier, Grenoble)
Abstract
We are going to discuss the following problem: to which extent the group of automorphisms of an affine algebraic variety determines
the variety? In general the answer is negative. On the other hand, H. Kraft proved that the group of automorphisms of the affine n-space
seen as an ind-group determines the affine n-space in the category of connected affine varieties.
In this talk we are going to discuss a similar result for affine toric varieties.
In case of dimension two, we characterise a big class of affine surfaces by their automorphism groups viewed as abstract groups.
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Isomorphisms of Chen-Ruan cohomology groups between different
GIT quotients
Yizhen ZHAO (Paris)
Abstract
Consider a torus action on a vector space $V$, we can consider diferent linearisations of the trivial line bundle on $V$
to get different quotients in Geometric Invariant Theory (GIT). If the action satisfies the so-called Calabi-Yau condition,
then we can show that different GIT quotients leads to the same Chen-Ruan cohomology groups as graded vector spaces.
This observation has not been proved in general. I will give some examples.
Schedule
| Thursday |
Friday |
|
PELLETIER
09:00-10:00
|
ZHAO
10:15-11:15
|
POL
11:30-12:30
|
BIGNALET-CAZALET
11:30-12:30
|
| Lunch |
|
ACHET
13:45-14:45
|
REGETA
15:00-16:00
|
PAL
16:30-17:30
|
18:30 Social Picnic
|
Practical Informations
Organizers
Adrien Dubouloz, Ronan Terpereau (Dijon)
Sponsors
