# North German Algebraic Geometry Seminar WS 2011/12

## Schedule

Thursday 10.11.2011(lecture hall 1208-a001 (Kesselhaus), Schloßwender Straße 5) | ||

14:15 - 15:15 h | Nicola Tarasca | Brill-Noether loci in codimension two |

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

15:30 - 16:30 h | Marc Nieper-Wißkirchen | Hochschild cohomology of compact Kähler manifolds and applications especially to holomorphic symplectic manifolds |

16:30 - 16:45 h | Coffee break | |

16:45 - 17:45 h | Osmanbey Uzunkol | Class invariants, optimality and related problems |

18:30 h | Conference Dinner at restaurant bei Mario, Schloßstr. 6 |

Friday 11.11.2011(lecture hall 1101-b305) | ||

09:30 - 10:30 h | Lutz Hille | Tilting Bundles on Rational Surfaces |

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

11:00 - 12:00 h | Jan Steffen Müller | Applications of dual graphs in arithmetic geometry |

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

13:30 - 14:30 h | Victoria Hoskins | Harder-Narasimhan stratifications |

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

15:00 - 16:00 h | Nicola Pagani | On moduli of bielliptic curves |

## Abstracts

**Tilting Bundles on Rational Surfaces**

Lutz Hille (Münster)

**Abstract**: It is a classical problem to construct derived equivalences between coherent sheaves on (smooth projective) algebraic varieties and modules over finite dimensional algebras. Such an equivalence is given by a tilting object. In this talk we are mainly interested in algebraic surfaces and tilting bundles. We consider a smooth projective rational surface and construct tilting bundles on it. In my talk I review the joint results with Markus Perling on tilting bundles on toric surfaces and its consequences for rational surfaces. Then I describe the finite dimensional endomorphism algebras of the tilting bundles and explain toric systems. In the last part I give an overview on recent results with David Ploog on certain subcategories that are invariant under spherical twists.

**Harder-Narasimhan stratifications **

Victoria Hoskins (Oxford, GBR/Freie Universität Berlin)

**Abstract**: When a reductive group acts on a complex projective scheme with respect to an ample linearisation the geometric invariant theory (GIT) quotient provides a categorical quotient of an open subscheme of this projective scheme, known as the GIT semistable subscheme. Associated to this action there is a stratification of the complex projective scheme such that the GIT semistable subscheme is the open stratum. We consider the problem of how to construct quotients of the group action on the unstable strata and provide a categorical quotient of each unstable stratum. We then study a stratification of a quot scheme parameterising sheaves over a projective scheme which has an action by a special linear group. We relate this stratification with a stratification by Harder-Narasimhan type and describe the categorical quotient of a Harder-Narasimhan stratum. We describe why this quotient is not a suitable quotient of the Harder-Narasimhan stratum and then discuss how we can rectify this problem.

**Applications of dual graphs in arithmetic geometry **

Jan Steffen Müller (Universität Hamburg)

**Abstract**: Let *X* be a smooth projective curve defined over a number field *K* and let X be a semistable regular model of of *X* over Spec (*O*_{K}). Then one can metrize and do harmonic analysis on the dual graphs Γ_{p} of the special fibers of X at non-archimedean primes **p** of *O*_{K}.

In this talk we will first discuss how this can be used to obtain lower bounds for the Arakelov-self-intersection of the relative dualizing sheaf of X (originally due to Zhang, joint work in progress with Kühn). Then we show that certain Néron functions with respect to **p** on the Jacobian *J* of *X* can be computed essentially on Γ_{p} (joint work with Stoll). This can be used, for instance, to compute Néron-Tate heights on *J*.

**Hochschild cohomology of compact Kähler manifolds and applications especially to holomorphic symplectic manifolds **

Marc Nieper-Wißkirchen (Universität Augsburg)

**Abstract**: The Hochschild cohomology ring and the module of Hochschild homology over it are invariants of the derived category of a compact Kähler manifold *X*. The Hochschild cohomology is isomorphic as a ring to the cohomology ring of poly-vector fields on *X* and Hochschild homology is isomorphic as a module to the (suitably graded) Dolbeault cohomology of *X*. Hochschild cohomology is also a natural domain for the Chern character of a bounded complex of coherent sheaves on *X*.

In the talk we will present a number of implications of these results for the theory of holomorphic symplectic manifolds.

** On moduli of bielliptic curves **

Nicola Pagani (Leibniz Universität Hannover)

**Abstract**: Moduli spaces of covers of curves have been of classical and more recent interests in the field of moduli spaces. One can investigate the geometry of these moduli spaces or, in a different direction, study the class of those curves in *M _{g}* that are covers of degree

*d*of a curve of a lower genus

*g'*(

*d*and

*g'*fixed). Both the questions have been thoroughly studied in the case when

*g'*equals 0. In this talk, we discuss the simplest next case: namely the case of moduli spaces of bielliptic curves. We present two kind of results:

- The computation of the Picard group and on the Kodaira dimension of the moduli spaces of bielliptic curves. These results are extended to the more natural setting of moduli of cyclic covers of stable genus 1 curves.
- We discuss a geometric construction that allows the computation (a joint work with C. Faber) of the class of the bielliptic locus in
*M*in terms of boundary strata, or of other tautological classes._{3}

**Brill-Noether loci in codimension two **

Nicola Tarasca (Humboldt-Universität Berlin)

**Abstract**: In recent years, divisors of moduli spaces of curves have been extensively studied. Computing classes in codimension one has yielded important results on the birational geometry of the moduli spaces of curves. On the contrary, classes in codimension two are basically unexplored.

In the moduli space of curves of genus 2k we consider the locus defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we will show how to compute the class of its closure in the moduli space of stable curves.

** Class invariants, optimality and related problems **

Osmanbey Uzunkol (Carl von Ossietzky Universität Oldenburg)

**Abstract**: Construction of Jacobians of elliptic and hyperelliptic curves with given number of points over a finite field have been playing important roles in various fields, such as primality proving, curve and pairing based cryptography.

This problem has also a relation to the famous 12. Problem of Hilbert. Using the theory of complex multiplication and suitable generalizations of class invariants of Weber as quotients of values of "Thetanullwerte", one can obtain smaller generators of some class fields which makes it possible to construct effectively the curves, whose Jacobians have given number of points.

In this talk the construction of class invariants with "Thetanullwerte" is explained together with the question of how optimal these class invariants are.

Moreover, the proof that most of the invariants introduced by Weber are actually units in the corresponding ring class fields will be given, which allows to obtain better class invariants in some cases, and to give an algorithm that computes the unit group of corresponding ring class fields