#### Institut für Mathematik

# North German Algebraic Geometry Seminar

WS 2010/2011

## Schedule

Thursday, 18-Nov-2010: (Lecture Hall W1 0-015) |
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14:30 - 15:20 | Chris Brav, LU Hannover | Braid groups and Kleinian singularities |

15:20 - 16:00 | Coffee Break | |

16:00 - 16:50 | Jeroen Sijsling, U Utrecht | Canonical models for arithmetic (1;e)-curves |

16:50 - 17:10 | Coffee Break | |

17:10 - 18:00 | Cristina Manolache, HU Berlin | Virtual push-forwards |

19:00 | Dinner at Restaurant "Tafelfreuden" |

Friday, 19-Nov-2010:((Lecture Hall W1 0-015) |
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09:50 - 10:40 | Michael Stoll, U Bayreuth | Torsion points on elliptic curves over quartic number fields |

10:40 - 11:10 | Coffee Break | |

11:10 - 12:00 | Hans-Christian Graf von Bothmer, U Göttingen | Affine Groups and Rationality |

12:00 - 13:50 | Lunch Break | |

13:50 - 14:40 | Elena Martinengo, FU Berlin | A new perspective on deformations of complex manifolds |

14:40 - 14:50 | Coffee Break | |

14:50 - 15:40 | Max Pumperla, U Hamburg | A tropical view on Landau-Ginzburg models |

### Abstracts

**Canonical models for arithmetic (1;e)-curves**

Jeroen Sijsling (Universität Utrecht)

**Abstract:**

A (1;e)-curve is a quotient of the upper half plane that is of genus 1 and ramifies above only one point. We explore the finite list, due to Kisao Takeuchi, of arithmetic (1;e)-curves, which are those (1;e)-curves that allow a natural finite-to-one correspondence with a Shimura curve coming from a quaternion algebra over a totally real field. After defining the notion of a canonical model for such an arithmetic (1;e)-curve, we show how to calculate these canonical models by using explicit methods such as p-adic uniformizations and Belyi maps along with modular techniques involving the Shimura congruence relation and Hilbert modular forms.**Braid groups and Kleinian singularities**

Chris Brav (LU Hannover)

**Abstract:**

We review the relation between the geometry of Kleinian singularities and Dynkin diagrams of types ADE, recalling in particular the construction of a braid group action of type A, D, or E on the derived category of coherent sheaves on the minimal resolution of a Kleinian singularity. By work of Seidel-Thomas, this action was known to be faithful in type A. We extend this faithfulness result to types ADE, which provides the missing ingredient for completing Bridgeland?s description of spaces of stability conditions for certain triangulated categories associated to Kleinian singularities. This is joint work with Hugh Thomas from the University of New Brunswick.**Virtual push-forwards**

Cristina Manolache (HU Berlin)

**Abstract:**

For a given a variety X, one can compactify the space of curves inside X in several ways. If the compactified space is "nice" enough (i.e. if it admits a "virtual class"), then one can define invariants. These are numbers which virtually count curves in X with certain prescribed properties. The goal of the talk is to give a method of comparing "curve-counting invariants". We will illustrate it by presenting a few examples in which we compare certain "virtual classes".**A new perspective on deformations of complex manifolds**

Elena Martinengo (FU Berlin)

**Abstract:**

We will start briefly recalling basic tools of classical deformation theory and some advantages of using differential graded Lie algebras and L-infinity algebras in it. In particular, we will concentrate on the analysis of deformations of complex manifolds and on a reinterpretation of Kodaira-Spencer approach by means of the modern language of stacks. We will review recent Getzler and Hinich’s results in this direction and present our refinement of them, obtained in a joint work with D. Fiorenza and M. Manetti. As further development, we will briefly explain how these tecniques can be applied to study deformations of a singular variety; this is part of a work in progress with D. Fiorenza and D. Iacono.**A tropical view on Landau-Ginzburg models**

Max Pumperla (Universität Hamburg)

**Abstract:**

The aim of this talk is to give an introduction to the study of Landau-Ginzburg models within the mirror symmetry program proposed by Gross and Siebert. The mirror of a Fano variety, a so called Landau-Ginzburg model, consists of a noncompact Kähler manifold with a holomorphic function called the superpotential. The existing literature mostly deals with toric Fano manifolds, where an explicit mirror construction is known. However, there are only a few ad hoc constructions available beyond the toric case. Given a toric degeneration of Fano manifolds we immediately obtain the Kähler manifold in question. To construct the superpotential, we use the technique of broken lines introduced by Gross. In this way we not only obtain a (partial) compactification of the mirrors known from the toric situation, but can also construct mirrors for certain non-toric varieties. We will illustrate the computability of our approach in the case of del Pezzo surfaces. If time permits, we will comment on enumerative aspects and the relation between broken lines and tropical disks.**Affine Groups and Rationality**

Hans-Christian Graf von Bothmer (Universität Goettingen)

**Abstract:**

In this talk I will describe two results obtained in joint work with Christian Boehning and Fedor Bogomolov. Fristly we can show that V/SAff(n) is rational for almost free irreducible SAff(n) representations of large dimension. Secondly we use representations of affine groups to show that V/SL(n) is stably rational of level n. Previosly only level n^2-1 was known in this generality.**Torsion points on elliptic curves over quartic number fields**

Michael Stoll (Universität Bayreuth)

**Abstract:**

Merel has shown that if we ﬁx d ≥ 1, then there are only ﬁnitely many group structures occurring as E(K)_{tors}, where E/K is an elliptic curve over a number ﬁeld of degree ≤ d. The crucial step is to show that the set S(d) of possible prime orders of torsion points on such curves is ﬁnite. It is then a natural question to ask for an explicit description of S(d). It is known that S(1) = {2, 3, 5, 7} and S(2) = S(3) = {2, 3, 5, 7, 11, 13}. In joint work with Sheldon Kamienny and William Stein, we have succeeded in showing that S(4) = {2, 3, 5, 7, 11, 13, 17}. I will give an overview of the proof.