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Relative Chow-Künneth decompositions for quadric bundles
Mattia CAVICCHI (IRMA Strasbourg)
Abstract
Let $k$ be an algebraically closed field of characteristic zero or the separable closure of a finite field. The decomposition theorem of
Beilinson-Bernstein-Deligne-Gabber says that for any proper morphism $f:X\rightarrow S$ of varieties over $k$, with $X$ smooth over $k$,
the total direct image along $f$ of the constant local system $\mathbb{Q}_X$ (when $k$ embeds into $\mathbb{C}$) or of the constant
$\ell$-adic sheaf $\mathbb{Q}_{\ell,X}$ ($\ell$ any prime different from the characteristic of $k$) 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 joint work with F. Déglise and J. Nagel, one of whose consequences is
a proof of the relative Chow-Künneth conjecture when $f:X\rightarrow S$ is a "general" quadric bundle.
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Connected algebraic groups acting on surfaces
Pascal FONG (Basel)
Abstract
We classify the maximal connected algebraic subgroups of $\mathrm{Bir}(S)$, when $S$ is a surface.
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Moduli spaces of quasitrivial rank 2 sheaves
Douglas GUIMARÃES (IMB)
Abstract
A torsion free sheaf $E$ on $\mathbb{P}^3$ is called quasitrivial if $E^{\vee\vee}=\mathcal{O}_{\mathbb{P}^3}^{\oplus r}$ and $\dim(E^{\vee\vee}/E)=0$.
While such sheaves are always -semistable, they may not be Gieseker semistable. We study the moduli spaces of - and Gieseker semistable quasitrivial sheaves of
rank 2 via the quot scheme of points $Quot(\mathcal(O)_{\mathbb{P}^3}^{\oplus 2},n)$, where $n=h^0(E^{\vee\vee}/E)$. We will show the construction of an
irreducible component of the Gieseker moduli space which is birrational to the total space of a $\mathbb{P}^{n-1}$-bundle over $S(n-1)\times\mathbb{P}^3$,
where $S(n)$ is the smoothable component of the Hilbert scheme of $n$ points in $ \mathbb{P}^3$. Furthermore, this is the only irreducible component when $n\le10$.
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On the parabolic subgroups associated to a reductive group in positive characteristic
Marion JEANNIN (ICJ Lyon)
Abstract
This talk aims to present results obtained during the speaker’s PhD and that partially answerthe following question: given a reductive group scheme $G$
over a curve $X$ defined over a field $k$,what should the canonical parabolic subgroup associated to $G$ be? This is actually a key point to
extend the notion of (semi-)stability, already defined for points by geometric invariant theory, to reductive group schemes.
Several generalisations exist depending on the context and all of them associate to $G$ a parabolic subgroup. When $k$ is of characteristic $0$, if
these subgroups coexist (which depends on hypotheses on $G$ and $k$), they are the same. In characteristic $p >0$ the situation complexifies.
Comparisons of the various candidates in this last setting require to obtain an analogue of Morozov theorem (which characterises parabolic Lie subalgebras of Lie($G$)
by means of their nilradical) in positive characteristic.
This last problem has recently been answered by V. Balaji, P. Deligne and A. J. Parameswaran first, with a uniform approach which unfortunately requires mild
conditions on $p$ and $G$, then by A. Premet and D. I. Stewart, with a case-by-case proof which completely breaks down the needed hypotheses on $p$ and $G$.
We will present in this talk results that adapt V. Balaji, P. Deligne and A. J. Parameswaran tools to approach the level of generalities of A. Premet and D. I. Stewart.
If time permits, we will discuss how the obtained analogue allows to compare the different canonical parabolic subgroups previously mentioned.
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Equivariant Cox ring of an algebraic variety with algebraic 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, and following
a classical method in invariant theory, I will give a description by generators and relations of 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 Cox ring, and why it is an interesting question.