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 LelliChiesa (HU Berlin)
15:15–16:15 Geom H2: HansChristian Graf v. Bothmer (Universität Hamburg)
Abstracts:
 Finite Field Experiments and the Poincare Center Problem
HansChristian 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 wellknown 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 socalled 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 rank3 LazarsfeldMukai bundles on K3 surfaces
Margherita LelliChiesa (HU Berlin)
Abstract:
LazarsfeldMukai bundles have proved useful in BrillNoether
theory of curves on K3 surfaces. Given an ample line bundle L on a K3
surface S, we study the Lslope stability of rank 3 LazarsfeldMukai
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 BrillNoether
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 BrillNoether theory
and transversality of BrillNoether 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 nonconstant 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 Ktheory 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 wellunderstood and can be described
through localization in terms of ``piecewise Laurent polynomials";
this is the Ktheory 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 Ktheory" 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 ndimensional 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 MiróRoig.
 Homological mirror symmetry for varieties of general type
Helge Ruddat (Universität Mainz)
Abstract:
We describe a mirror construction of LandauGinzburg 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 noncommutative Hodge structures. As an example, we give 3 different constructions to produce a mirror of a genus two curve.
