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Inhomogeneous Kleinian singularities and quivers
Antoine CARADOT (Lyon)
Abstract
In 1998 H. Cassens and P. Slodowy developed a construction of the semiuniversal deformations
of the simple singularities of types $A_r, D_r, E_6, E_7$ and $E_8$. In this talk we will present
a generalization of this construction to the singularities of inhomogeneous types $B_r, C_r, F_4$ and $G_2$.
By studying the representation space of a quiver defined from a simple singularity $\mathbb{C}^2 / \Gamma$ of
homogeneous type $\Delta(\Gamma)$ via the McKay correspondence, and equipped with the action of a well
chosen finite subgroup $\Gamma'$ of $SU_2$ containing $\Gamma$ as normal subgroup, we will use the symmetry
group $\Omega = \Gamma' / \Gamma$ of the Dynkin diagram $\Delta(\Gamma)$ and explicitly compute
the semiuniversal deformation of the singularity $(\mathbb{C}^2 / \Gamma, \Omega)$ of inhomogeneous type,
whose type is linked to the folding of the Dynkin diagram $\Delta(\Gamma)$.
The fibers of this deformation are all equipped with an induced $\Omega$-action. By quotienting we obtain
a deformation of the singularity $\mathbb{C}^2 / \Gamma'$ with some unexpected fibers. We will give examples
with the types $D_4$ as well as $A_{2r-1}$ with small ranks.
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Unstable points for torus actions on flag varieties
Benoît DEJONCHEERE (Lyon)
Abstract
The aim of this talk is, under reasonable assumptions, a better understanding of (complex)
torus actions on flag varieties $G/P$. We will investigate these actions through
Geometric Invariant Theory, and we will give an expression of the set of unstable points
relatively to an ample line bundle linearized by the given torus, by starting with the case
of maximal tori acting on full flag varieties $G/B$. In a second time, we will look at the case
when $P$ is a maximal parabolic subgroup of $G$, and we will get a a few consequences of this description.
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Isomorphisms between complements of unicuspidal curves in the projective plane
Mattias HEMMIG (Basel)
Abstract
In 2012 Costa constructed a family of unicuspidal curves in the projective plane
that are pairwise non-equivalent but have isomorphic complements. We call a line
$L$ \empph{very tangent} to a projective plane curve $C$ if $L$ and $C$ intersect in only one point.
We show that a family as the one of Costa does not exist for a unicuspidal curve $C$ that admits a very
tangent line through the singular point. Such a curve corresponds to to the closure in the projective plane
of an embedded affine line in the affine plane. To state the result more precisely, if $D$ is any plane curve
and there exists an isomorphism between the complements of $C$ and $D$, then the two curves are projectively e
quivalent, even though the isomorphism is not necessarily linear. The proof works over an algebraically closed
field of any characteristic and generalizes a result of Yoshihara (1984) who proved the claim over the complex numbers.
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Cylinders in Mori Fiber Spaces
Takashi KISHIMOTO (Saitama, Japan)
Abstract
In consideration of an application about existence of unipotent group actions on affine cones, it is often very important to look into
cylinders found on projective varieties. In some sense, it is essential for us to concentrate on cylinders contained in Mori Fiber Space
$\pi: V \to W$, indeed, an existence of a cylinder respecting the structure of $\pi$ is translated into an $\mathbb{A}_K^1$-cylinder
in the generic fiber $V_\eta$ of $\pi$, where $K=\mathbb{C} (\eta) =\mathbb{C} (W)$ is the function field of $W$.
Let $r= \mathrm{dim}(V) - \mathrm{dim}(W)$ be the relative dimension. In case of $r=1$ (Mori conic bundle), it is easy to see that
$V_\eta$ contains an $\mathbb{A}_K^1$-cylinder if and only if $V_\eta$ admits a $K$-rational point.
Whereas, for $r \geqq 2$, the observation becomes to be quite complicated even if we assume that $V_\eta$ possesses $K$-rational points.
In this talk, we devote ourselves mainly to the case of $r=2$ and $r=3$.
For $r=2$ (del Pezzo fibration), we shall give a complete criterion about existence of an $\mathbb{A}_K^1$-cylinder on $V_\eta$ depending
on the degree of $V_\eta$. Further, for $r=3$ also, we obtain an criterion provided that $V_\eta$ is a $K$-form of a smooth Fano
threefold with Fano index bigger than one. This is a joint work with Adrien Dubouloz.
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Branes et symétrie miroir sur les variétés hyperkählériennes
Grégoire MENET (Dijon)
Abstract
Un brane est un objet mathématique venant de la physique, en particulier
de la théorie des cordes. Sur les variétés hyperkählériennes, il s'agit
d'une sous-variété qui est soit complexe soit lagrangienne selon chacune
des trois structures complexes ambiantes. Dans cet exposé, nous étudierons
comment les branes obtenus comme lieux fixes d'involutions évoluent sous
l'action de la symétrie miroir.
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Codimension two torus actions and linearisation conjecture
Charlie PETIJEAN (Dijon)
Abstract
An algebraic torus action on the affine space is it conjugated to a linear one in the automorphisms group ?
We will recall the results obtained in the 90s about specific case where the torus is 1-dimensional and affine space is 3-dimensional.
This in a more recent and adapted formalism to a generalization, ie, the language of p-divisors developed by Altmann and Hausen.
In a second step, we will present joint work in progress with A. Liendo (U. Talca) concerning codimension two torus action in any dimension.
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Singularities of type $A_k$, Newton polygons and Hirzebruch surfaces
Julia SCHNEIDER (Basel)
Abstract
One can look at the curves on the projective plane of degree $d$ and ask: What is the largest singularity
of type $A_k$ that such a curve can have? For small degrees, this is known. Starting from degree 7, the answer
seems to be unknown and asymptotic behaviours are also open, even if bounds are known. We will look at
polynomials on the affine plane with a certain Newton polygon and will give answers for ``small'' triangles
using Hirzebruch surfaces, and birational maps between these.
Adrien DUBOULOZ 
