# Summer Workshop in Algebraic & Arithmetic Geometry

This workshop covers problems in arithmetic and algebraic geometry that are related to torsors, Cox rings and rational points, using abstract and explicit methods.

Information

- Dates: 5/6 July 2012 (Thursday/Friday)
- Location: Room B 349, Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München (travel information)
- Organizers: Ulrich Derenthal (LMU), Jürgen Hausen (Tübingen)
- Supported by: Center for Advanced Studies (LMU), DFG
- Poster (PDF)

Invited talks

- Cyril Demarche (Paris 6)
- Christopher Frei (LMU)
- Alex Küronya (Budapest / Freiburg)
- Antonio Laface (Concepción)
- Vladimir Lazić (Bayreuth)
- Anthony Várilly-Alvarado (Houston)
- Dasheng Wei (Beijing / LMU)

Student talks

- Hendrik Bäker (Tübingen)
- Elaine Herppich (Tübingen)
- Simon Keicher (Tübingen)

Schedule

Thursday, 5 July 2012 (room B 349) | |
---|---|

9:10-10:00 | Alex Küronya: Okounkov bodies and rationality of volumes of line bundles |

Coffee | |

10:30-11:00 | Elaine Herppich: Almost homogeneous Del Pezzo C^{*}-surfaces |

11:10-12:00 | Christopher Frei: Counting rational points over number fields on a cubic surface |

Lunch break | |

14:40-15:30 | Anthony Várilly-Alvarado: Failure of the Hasse principle on general K3 surfaces |

Coffee | |

16:10-16:40 | Simon Keicher: Computing Cox rings via ambient modifications |

16:50-17:40 | Vladimir Lazić: On the Morrison-Kawamata cone conjecture |

Friday, 6 July 2012 (room B 349) | |

10:00-10:50 | Cyril Demarche: Unramified Brauer group of homogeneous spaces |

Coffee | |

11:20-12:10 | Dasheng Wei: Universal torsors and values of quadratic polynomials represented by norms |

Lunch break | |

14:10-14:40 | Hendrik Bäker: Chow quotients of torus actions |

Coffee | |

15:10-16:00 | Antonio Laface: Elliptic fibrations from cubic threefolds |

Abstracts

**Hendrik Bäker:***Chow quotients of torus actions*

Let an algebraic group G act on a normal projective variety X. For this action we consider the Chow quotient Y, which in general turns out to be quite hard to access. However, in the case where G is an algebraic torus the Chow quotient is (up to normalisation) isomorphic to a closed subset of a toric variety. We now ask, if the Cox ring of the Chow quotient Y is finitely generated provided the Cox ring of X is so. For toric varieties and those with a complexity-1 torus action the answer is positive. In general, however, this is an open problem; for example it is unknown whether M(0,n), which arises as Chow quotient of the Grassmannian, has a finitely generated Cox ring when n>6. Using tropical and convex geometry we provide an approach for studying finite generation in some cases. In particular we prove that the Chow quotient of projective quadrics with respect to a K*-action always has this property.**Cyril Demarche:***Unramified Brauer group of homogeneous spaces*

Let K be a field and let X be a homogeneous space of a connected linear algebraic group G over K. The unramified Brauer group of X is a cohomological invariant of X that plays an important role in the arithmetic of X (when K is a global field, via the Brauer-Manin obstruction to weak approximation) and in the geometry of X (especially to study the rationality of X over arbitrary fields K). In this talk, we will show a formula for the unramified Brauer group, under some connectedness assumption for the stabilizers of points in X. In particular, we will focus on the case of algebraically closed fields, global fields and finite fields. This is joint work with Mikhail Borovoi and David Harari.**Christopher Frei:***Counting rational points over number fields on a cubic surface*

Let K be a number field. A conjecture of Manin predicts the asymptotic distribution of K-rational points of bounded height on certain algebraic varieties defined over K. In recent years, an approach using universal torsors has been applied to prove several hard special cases of Manin's conjecture when K is the field of rational numbers. It is of natural interest to generalize this method to arbitrary number fields. We discuss a first step in this direction, a proof of Manin's conjecture for the singular cubic surface defined by W^{3}= X Y Z over any number field, using universal torsors.**Simon Keicher:***Computing Cox rings via ambient modifications*

We present a technique to compute Cox rings using toric ambient modifications. As examples, we consider certain blow ups of P^{2}.**Antonio Laface:***Elliptic fibrations from cubic threefolds*

Let Y be a smooth projective cubic threefold and let L be a line of P^{4}not contained in Y. The minimal resolution of the projection of Y from L is an elliptic fibration f: X -> P^{2}defined on a threefold X. In this talk I will describe the Mordell-Weil groups of such fibrations and prove, following a recent work of A. Prendergast-Smith about extremal rational elliptic threefolds, that X is a Mori dream space if and only if the Mordell-Weil group of f is finite. This is work in progress joint with Andrea Tironi and Luca Ugaglia.**Vladimir Lazić:***On the Morrison-Kawamata cone conjecture*

If X is a Calabi-Yau manifold, then an important conjecture by Morrison and Kawamata predicts that the nef and mobile cones have a particularly nice desription in terms of actions of the groups Aut(X) and Bir(X). The conjecture has been verified for surfaces, and only in a small number of higher dimensional cases. In this talk, I will show how a certain version of the conjecture is true on Calabi-Yau 3-folds with Picard number 2, using a recent result of Oguiso. This is a joint work in progress with Thomas Peternell.**Anthony Várilly-Alvarado:***Failure of the Hasse principle on general K3 surfaces*

Transcendental elements of the Brauer group of an algebraic variety, i.e., Brauer classes that remain nontrivial after extending the ground field to an algebraic closure, are quite mysterious from an arithmetic point of view. These classes do not arise for curves or surfaces of negative Kodaira dimension. In 1996, Harari constructed the a 3-fold with a transcendental Brauer-Manin obstruction to the Hasse principle. Until recently, his example was the only one of its kind. We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class α that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X,α) is constructed from a double cover of P^{2}x P^{2}ramified over a hypersurface of bi-degree (2, 2). This is joint work with Brendan Hassett.**Dasheng Wei:***Universal torsors and values of quadratic polynomials represented by norms*

Let K/k be an extension of number fields, and let P(t) be a quadratic polynomial over k. Let X be the affine variety defined by P(t) = N_{K/k}(z). We study the Hasse principle and weak approximation for X in two cases. For [K:k]=4 and P(t) irreducible over k and split in K, we prove the Hasse principle and weak approximation. For k=**Q**with arbitrary K, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one.

Participants

- Ivan Arzhantsev (LMU / Moscow)
- Hendrik Bäker (Tübingen)
- Sabine Bauer (LMU)
- Cyril Demarche (Paris 6)
- Ulrich Derenthal (LMU)
- Christopher Frei (LMU)
- Fabian Gundlach (LMU)
- Markus Hanselmann (LMU)
- Jürgen Hausen (Tübingen)
- Elaine Herppich (Tübingen)
- Simon Keicher (Tübingen)
- Tamäs Korodi (LMU)
- Alex Küronya (Budapest / Freiburg)
- Antonio Laface (Concepción)
- Vladimir Lazić (Bayreuth)
- Felix Rötting (TUM)
- Christoph Schießl (LMU)
- Anthony Várilly-Alvarado (Houston)
- Dasheng Wei (Beijing / LMU)
- Wolfgang Zehentner (LMU)