Arithmetic and geometry of cyclic covers
When presented with a hypersurface in projective space, a common
trick is to pass to a cyclic cover of the projective space
branched along the given hypersurface. In this talk I will
discuss various stack-theoretic issues which arise when trying
to perform this construction in a family, and give applications
to finiteness results for hypersurfaces over $Q$ with good
reduction outside a given set of primes. This is joint work with
Ariyan Javanpeykar and Siddharth Mathur.
Marta Pieropan
Campana points on toric varieties
We call Campana
points an arithmetic notion of points on Campana's orbifolds
that has been first studied by Campana and Abramovich,
and that interpolates between the notions of rational and
integral points. This talk will
introduce Campana points on toric varieties, present
geometric heuristic arguments in support of their
conjectured distribution and the currently known
asymptotic results. This is joint work with Damaris
Schindler.
Stefan Schreieder
The diagonal of quartic fivefolds
We show that a very
general quartic hypersurface in $\mathbb{P}^6$ over a field of characteristic different from $2$
does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise--Ottem, who showed stable irrationality over fields of characteristic zero. To prove our result, we introduce a new cycle-theoretic obstruction that may be seen as an analogue of the motivic obstruction for
rationality in characteristic
zero, introduced by Nicaise--Shinder and
Kontsevich--Tschinkel. This is joint work with Nebojsa Pavic.
Vadim Vologodsky
Quantizations of the category of coherent
sheaves on symplectic varieties
I will explain a construction due to Roman
Travkin of a canonical quantization ofthe category of coherent
sheaves on symplectic varieties over the ring of integers.