# North German Algebraic Geometry Seminar SS 2013

## Schedule

Thursday 16.5.2013(lecture hall 1101-b305) | ||

14:15 - 15:15 h | Wenfei Liu (Bielefeld) | Automorphisms of irregular surfaces of general type acting trivially in cohomology |

15:15 - 15:45 h | Coffee break | |

15:45 - 16:45 h | Ulrich Derenthal (LMU München) | Cox rings and rational points on del Pezzo surfaces |

16:45 - 17:00 h | Coffee break | |

17:00 - 18:00 h | Rene Pannekoek (Leiden) | On p-adic density of rational points on K3 surfaces |

18:30 h | Conference Dinner at restaurant Mikado, Schmiedestr. 3 |

Friday 17.05.2013(lecture hall 1101-b305) | ||

09:30 - 10:30 h | Stefan Keil (HU Berlin) | Non-square order Tate-Shafarevich groups of non-simple abelian surfaces |

10:30 - 11:00 h | Coffee break | |

11:00 - 12:00 h | Robin de Jong (Leiden) | Asymptotics of the Néron height pairing and a conjecture of Hain |

12:00 - 13:30 h | Lunch break | |

13:30 - 14:30 h | Kay Rülling (FU Berlin) | K-groups of Reciprocity functors |

14:30 - 15:00 h | Coffee break | |

15:00 - 16:00 h | Sabir Gusein-Zade (Moscow/LUH) | Higher order generalized Euler characteristics and generating series |

## Abstracts

**Cox rings and rational points on del Pezzo surfaces**

Ulrich Derenthal (LMU München)

For del Pezzo surfaces over number fields, the distribution of

rational points is predicted by Manin's conjecture. This has been

extensively studied over the field Q of rational numbers using

universal torsors and Cox rings. In this talk, I will give an

introduction to this subject and present joint work with Christopher

Frei generalizing this to imaginary quadratic number fields.

**Higher order generalized Euler characteristics and generating series**

S.M.Gusein-Zade (Moscow/LUH)

For a complex quasi-projective manifold with a finite group action, higher order Euler characteristics are generalizations of the orbifold Euler characteristic introduced by physicists. The generating series of the higher order Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them were computed by H.Tamanoi. I'll discuss motivic versions of the higher order Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line and give formulae for the generating series of these generalized Euler characteristics for the wreath product actions.

**Asymptotics of the Néron height pairing and a conjecture of Hain**

Robin de Jong (Leiden)

The Néron height pairing is a canonical real-valued pairing between divisors of degree zero and disjoint support on a compact Riemann surface. It serves as an archimedean contribution to the global canonical (Néron-Tate) height pairing between points on the jacobian of a curve over a number field. In this talk we study the asymptotics of the Néron height pairing on a family of degenerating Riemann surfaces parametrized by an algebraic curve. The limit behavior has a rather simple form, which is explained by a non-archimedean (Q-valued) analogue of the Néron pairing. As an application of our limit formula we prove part of a conjecture on 'height jumping' formulated recently by R. Hain. Joint work with David Holmes.

**Non-square order Tate-Shafarevich groups of non-simple abelian surfaces**

Stefan Keil (HU Berlin)

For an elliptic curve (over a number field) it is known that the order of its Tate-Shafarevich group is a square, provided it is finite. In higher dimensions this no longer holds true. We will present work in progress on the classification of all occurring non-square parts of orders of Tate-Shafarevich groups of non-simple abelian surfaces over the rationals. We will prove that only finitely many cases can occur. To be precise only the cardinalities k=1,2,3,5,6,7,10,13,14,26 are possible. So far, for all but the last three cases we are able to show that these cases actually do occur by constructing explicit examples.

**Automorphisms of irregular surfaces of general type acting trivially in cohomology**

Wenfei Liu (Bielefeld)

Let X be a complex nonsingular variety, it is natural to consider the action of the automorphism group Aut(X) on the cohomology ring H^(X, R), where R is some coefficient ring. One wants to know the kernel of this action. In this talk I will report some results in the case of irregular surfaces of general type. This is joint work with Jin-Xing Cai, Lei Zhang and some additional joint work in progress with Jin-Xing Cai.

**On p-adic density of rational points on K3 surfaces**

Rene Pannekoek (Leiden)

For an elliptic curve E over the rational numbers, let X be the Kummer surface of E times E. This is an example of a K3 surface. For each prime number p, I will give conditions on E that imply X(Q) to be dense in the space X(Q_p) of p-adic points of X. I will also give examples of E such that X(Q) is dense in X(Q_p) for p in a set of primes with density 1/2. Lastly, I will discuss density results for product topologies.

**K-groups of Reciprocity functors**

Kay Rülling (FU Berlin)

We introduce reciprocity functors; examples are provided by smooth commutative algebraic groups, Kähler differentials and homotopy invariant Nisnevich sheaves with transfers. Then we will define the K-groups of a tuple of reciprocity functors and compute them in some cases. In particular we obtain a description of the absolute Kähler differentials of degree n of a characteristic zero field as the K-group attached to the tuple consiting of the additive group and n-times the multiplicative group. This is joint work with Florian Ivorra.