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Recent results on plane Cremona transformations
Alberto CALABRI (Ferrara)
Abstract
I will report on a joint work with my student Nguyen Giao about the classification of plane Cremona maps of small degree.
I will review some known properties of the quasi-projective variety which parametrizes plane Cremona maps of fixed degree (a joint work with Cinzia Bisi and Massimiliano Mella).
I will then address the question of determining which plane Cremona maps of small degree are limits of maps of higher degree (a joint work with Jérémy Blanc).
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Relative Chow-Künneth decompositions for quadric bundles
Mattia CAVICCHI (IMB)
Abstract
The decomposition theorem of Beilinson-Bernstein-Deligne-Gabber says that for any proper morphism $f:X\to S$ of complex varieties, with $X$ smooth over $\mathbb{C}$,
the total direct image along $f$ of the constant sheaf $\mathbb{Q}_X$ decomposes, in the derived category, as a direct sum of certain complexes called
"intersection complexes". The relative Chow-Künneth conjecture predicts in particular that the projectors on the factors of such a direct sum are
induced by relative algebraic correspondences over S. The aim of the talk is to explain this circle of ideas and to report on a work in progress with
F. Déglise and J. Nagel, aimed at proving the relative Chow-Künneth conjecture when $f:X \to S$ is a quadric bundle.
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On the canonical parabolic subgroup of a reductive group scheme
Marion JEANNIN (ICJ Lyon)
Abstract
Let $k$ be a field and $G$ a reductive group scheme over a smooth projective geometrically connected $k$-curve $X$. When $G$ is the twisted form of a constant
reductive group scheme and when $k$ is algebraically closed of characteristic $0$, M. Atiyah and R. Bott defined the canonical parabolic subgroup of $G$.
It is the parabolic subgroup whose Lie algebra is given by the term of the Harder--Narasimhan filtration of $\mathrm{Lie}(G)$ (seen as a vector bundle) whose quotient
has slope $0$. In his thesis K. A. Behrend extended this notion to any reductive group scheme over a curve defined over any field. His definition does not
make use of the Harder--Narasimhan filtration of $\mathrm{Lie}(G)$. V. B. Mehta and S. Subramanian then showed that M. Atiyah and R. Bott's definition still
makes sense when $k$ is of characteristic $p> 2 \dim(G)$ in the twisted form framework.
In this talk we propose a new proof of this statement which allows to characterise this canonical object as the instability parabolic of a given
nilpotent subalgebra of $\mathrm{Lie}(G)$. The reasoning requires an analogue of Morozov's theorem for nilpotent
Lie algebras and allows to provide a completely uniform proof of M. Atiyah and R. Bott's result, with no restriction on the characteristic.
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Sums of squares and varieties of minimal degree
Daniel PLAUMANN (Universität Dortmund)
Abstract
In 1888, Hilbert classified all pairs (n,2d) such that every nonnegative real form of degree 2d in n variables can be expressed as a sum of squares of
forms of degree d. In 2013, Blekherman, Smith and Velasco discovered that Hilbert’s results could be deduced in a completely different manner using another
result from around the same time, the classification of varieties of minimal degree in projective space due to del Pezzo and Bertini.
More generally, this leads to a study of real quadratic forms non-negative on a real projective variety.
We will explain this remarkable connection and point out some subsequent developments. (Partly based on joint work with G. Blekherman, L. Chua, R. Sinn, and C. Vinzant).
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Actions of Cremona groups on CAT(0) cube complexes
Christian URECH (EPFL)
Abstract
To a variety $X$ we can associate its group of birational transformations Bir($X$). Recently, in geometric group theory, actions of groups on CAT(0)
cube complexes have turned out to be a useful tool to study various groups. I will explain a natural construction of CAT(0) cube complexes
on which Bir($X$) acts by isometries and explain how we can deduce new and old group theoretical and dynamical results from this action. This is
joint work with Anne Lonjou.
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Equivariant Cox ring of an algebraic variety with group action
Antoine VÉZIER (IF Grenoble)
Abstract
Let $k$ be an algebraically closed field of characteristic zero. The Cox ring is a
rich invariant of an algebraic variety $X$ over $k$ satisfying natural conditions. After a
quick introduction to the subject, I will present and motivate a construction of an equivariant
analogue of the Cox ring when $X$ is endowed with an action of an algebraic group $G$ over $k$.
This is called the equivariant Cox ring Cox$^G(X)$, which carries a natural structure of graded
$G$-algebra. Assuming that $X$ is rational of complexity one with $G$ connected reductive of trivial
Picard group, and following a classical method in invariant theory, I will give a presentation by
generators and relations on the algebra of $U$-invariants of Cox$^G(X)$, where $U$ is the unipotent
part of a Borel subgroup of $G$. In a second time, I will specialize to the case of the equivariant
Cox ring of an almost homogeneous SL$_2$-threefold. I will explain how the result on Cox$^G(X)^U$
can be used to study the singularities of the singularities of the Cox ring, and why it is an interesting question.
Ronan TERPEREAU .