UHH > Faculties > MIN-Faculty > Mathematics > News > Events > North German Algebraic Geometry Seminar > NoGAGS 2012 STiNE |  KUS-Portal |

### Program

Thursday, 3 May 2012:

14:00–15:00 Geom H1: Michael Lönne (Universität Hannover)
Coffee break
15:30–16:30 Geom H1: Helge Ruddat (Universität Mainz)
Coffee break
17:00–18:00 Geom H1: Sam Payne (Yale/MPIM)

18:30 Conference dinner: Terzetto

Friday, 4 May 2012:

09:00–10:00 Geom H1: Juan Pons (FU Berlin)
Coffee break
11:00–12:00 Geom H3: Claus Diem (Universität Leipzig)
Lunch
14:00–15:00 Geom H2: Margherita Lelli-Chiesa (HU Berlin)
15:15–16:15 Geom H2: Hans-Christian Graf v. Bothmer (Universität Hamburg)

Abstracts:

• Finite Field Experiments and the Poincare Center Problem
Hans-Christian Graf v. Bothmer (Universität Hamburg)
Abstract:
In 1885 Poincare asked when a plane autonomous system has closed integral curve in the neighbourhood of the origin. In this case we say that the system has a center. Poincare showed that the parameter space of such systems is an algebraic variety. Even today this center variety is known only in some very special cases. In this talk I will describe Poincare's problem and report on some recent results we obtained using finite field experiments. (This is joint work with Jakob Kröker and Johannes Steiner).
• On the discrete logarithm problem in elliptic curves
Claus Diem (Universität Leipzig)
Abstract:
It is well-known that the classical discrete logarithm problem, that is, the problem to compute discrete logarithms in the multiplicative groups of prime fields, can be solved in subexponential expected time. The same is true for the discrete logarithm problem in the multiplicative groups of all finite fields.
The corresponding algorithms are based on the so-called index calculus method. (''Index'' is a classical name for discrete logarithm.) I want to show in the talk that this method can also be applied successfully to the discrete logarithm problem in elliptic curves over finite extension fields. In particular, I want to outline how one can obtain the following theorem.
Let $a, b$ be two fixed positive real numbers. Then the discrete logarithm problem over fields of the form $\mathbb{F}_{q^n}$ with a prime power $q$ and a natural number $n$ satisfying $a \cdot \log(q)^{1/3} \leq n \leq b \cdot \log(q)$ can be solved in an expected time of $e^{O(\log(q^n)^{3/4})}$.
• Stability of rank-3 Lazarsfeld-Mukai bundles on K3 surfaces
Margherita Lelli-Chiesa (HU Berlin)
Abstract:
Lazarsfeld-Mukai bundles have proved useful in Brill-Noether theory of curves on K3 surfaces. Given an ample line bundle L on a K3 surface S, we study the L-slope stability of rank 3 Lazarsfeld-Mukai bundles associated to nets of type g^2_d on curves C in the linear system defined by L. When d is large enough and C is general, we obtain a dimensional statement for the variety W^2_d(C). If the Brill-Noether number is negative, we prove that any g^2_d is contained in a linear series which is induced from a line bundle on S, as conjectured by Donagi and Morrison. Some applications towards higher rank Brill-Noether theory and transversality of Brill-Noether loci are then discussed.
• Connected components of Hurwitz spaces
Michael Lönne (Universität Hannover)
Abstract:
The notion of Hurwitz space generally applies to moduli spaces of curves together with a non-constant regular map onto another curve.
Reporting on joint work with Fabrizio Catanese and Fabio Perroni we consider the case of curves $C$ with an effective action of a finite group $G$ and quotient map $C\to C'= C/G$. Alternatively $C$ can be considered as a Galois $G$-cover of $C'$ and thus gives rise to obvious numerical invariants, the genus of the quotient and the ramification multiplicities. We give a refined invariant and with its help completely classify all irreducible components of curves with an effective action of a dihedral group.
We have a conjecture that our invariant suffices to classify connected components for arbitrary groups G asymptotically, that is, if the numerical invariants are sufficiently large in a suitable sense. The talk concludes with evidence and a practicable strategy for our claim.
• Operational K-theory and localization for toric varieties
Sam Payne (Yale/MPIM)
Abstract:
The Grothendieck rings of ordinary and equivariant vector bundles on a smooth complete toric variety are well-understood and can be described through localization in terms of piecewise Laurent polynomials"; this is the K-theory analogue of the standard description of the cohomology rings in terms of piecewise polynomials on fans. A satisfactory understanding of Grothendieck rings of vector bundles on singular toric varieties, however, remains out of reach.
I will discuss joint work with Dave Anderson exploring an ''operational equivariant K-theory" that agrees with the Grothendieck ring of equivariant vector bundles on a smooth variety with torus action and can be described in terms of localization and piecewise polynomials on an arbitrary singular toric variety.
• Representation type of rational ACM surface in $\mathbb{P}^4$
Juan Pons (FU Berlin)
Abstract:
Since the seminal result by Horrocks characterizing Arithmetically Cohen Macaulay (ACM) bundles on the projective n-dimensional space as those that completely split an important amount of research has been devoted to the study of ACM or, in particular, Ulrich vector bundles (ACM bundles with the maximal permitted number of global sections). In particular, the complexity of a given projective variety can be studied in terms of the dimension and number of families of indecomposable ACM sheaves that it supports, namely, its representation type.
In this talk we focus on constructing indecomposable Ulrich sheaves of rank 1 and 2 on a general linear standard determinantal variety of small codimension. Moreover we determine the representation type of any smooth rational ACM surface in $\mathbb{P}^4$ by constructing large families of simple Ulrich bundles of arbitrary rank. It turns out that, excluding the cubic scroll, all of them are of wild representation type.
This is joint work with Rosa Maria Miró-Roig.
• Homological mirror symmetry for varieties of general type
Helge Ruddat (Universität Mainz)
Abstract:
We describe a mirror construction of Landau-Ginzburg models for which we proved a duality of Hodge numbers in a joint work with Gross and Katzarkov. We discuss how the generalized homological mirror symmetry conjecture fits in this framework and discuss Hochschild homology, cohomology and non-commutative Hodge structures. As an example, we give 3 different constructions to produce a mirror of a genus two curve.

 Impress 2014-04-24, wwwmath (MJ)