Zeitplan

Stefan Kebekus (Freiburg)

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first examply of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.

This is joint work with Jorge Pereira (IMPA) and Arne Smeets (Nijmegen).

Angela Ortega (HU Berlin)

Given a finite morphism between smooth projective curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map.

It is a classical result that the Prym map for étale double coverings over curves of genus at least 7 is generically injective but never injective (Donagi's tetragonal construction).

We will show that, unlike the unramified case, the Prym map is injective for ramified double coverings over curves of genus at least 1 and ramified in at least 6 points.

This is a joint work with J.C. Naranjo.

Jochen Heinloth (Duisburg-Essen)

For moduli problems it is often not so hard to show that they define algebraic stacks, but these are often not separated. This can sometimes be resolved by showing that some open "semistable" part of the moduli problem admits proper coarse moduli spaces. While we have necessary and sufficient conditions that an open the "semistable" part needs to satisfy to have a proper quotient, much fewer techniques are known to find such open subsets.

In the talk I will try to explain how looking at this problem for algebraic stacks, gives an approach to the problem to characterize the open subsets of smooth projective varieties with an action of a torus that admit a proper quotient space.