NoGAGS Berlin 2012

Freie Universität Berlin December 06-07, 2012 - A joint seminar with Bremen, Humboldt University Berlin, FU Berlin, Groningen, Hamburg, and Hannover. (Other NoGAGS meetings can be found [here])

Organizers

Alexander Schmitt, Anna Wißdorf

Schedule

Time	Thursday, 06.12.	Friday, 07.12.
9:30		T. Finis (FU Berlin) "An approximation theorem for congruence subgroups"
10:30		Coffee
11:00		J. Kass (Hannover) "What is H_{1}(Abel map)? "
12:00		Lunch
14:00	S. Rollenske (Bielefeld) "Pluricanonical maps of stable surfaces"	P. Sosna (Hamburg) "On the Jordan-Hölder property for geometric derived categories "
15:00	Coffee	Coffee
15:30	F. Gounelas (HU Berlin) "Free curves on varieties"	F. Reede (Göttingen) "Line bundles on noncommutative surfaces"
16:30	Coffee
17:00	A. Anema (Groningen) "Covering spaces of an elliptic curve that ramify only above one point"
18:30	Dinner at Eierschale

All talks (60 min each) are at Großer Hörsaal in Zuse-Institut Berlin.

Abstracts

• A. Anema: "Covering spaces of an elliptic curve that ramify only above one point"

ABSTRACT: This talk deals with finite maps to elliptic curves E defined over the complex numbers. From algebraic topology and the theory of Riemann surfaces, one knows there exist curves D admitting a finite map g : D --> E such that g ramifies only above one point of E. We consider the problem of explicitly constructing such pairs (D, g). This is done by looking at torsion of the elliptic surface corresponding to y^2=x^3+ax+b over the curve E given by 4a^3+27b^2=1.

• T. Finis: "An approximation theorem for congruence subgroups"

ABSTRACT: By a classic theorem of Jordan (1878), every finite subgroup of GL (n, K), where K is a field of characteristic zero, contains an abelian normal subgroup of index at most J(n), where J(n) depends only on n. In characteristic p the situation is of course different. A theorem of Nori (1987) says that for all n > 0 and all primes p with p > N(n), where N is a suitable function, the subgroups of GL (n, F_p) which are generated by their elements of order p are described by connected algebraic subgroups of GL (n) defined over F_p. This result can be combined with Jordan's theorem to describe arbitrary subgroups (cf. also Larsen-Pink 2011).

Let G be a reductive algebraic group defined over Q. In the talk I will present an approximation theorem for subgroups of G (Z/p^N Z) (or, equivalently, for open subgroups of G (Z_p)), which provides a partial description of these subgroups in terms of connected algebraic subgroups of G defined over Q_p. The theorem has applications to the theory of congruence subgroups of arithmetic groups, in particular to the limit multiplicity problem. The results are joint work with Erez Lapid (Jerusalem/Rehovot).

• F. Gounelas: "Free curves on varieties"

ABSTRACT: We study various ways in which a variety can be "connected by curves of a fixed genus", mimicking the notion of rational connectedness. At least in characteristic zero, in the specific case of the existence of a single curve with a large unobstructed deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

• J. Kass: "What is H_{1}(Abel map)? "

ABSTRACT: A smooth curve over the complex numbers admits an Abel map, that is, an embedding into the complex torus known as the Jacobian, and the homomorphism on homology induced by the Abel map can be identified with the Poincaré Duality isomorphism. I will describe how this result extends to singular curves. In doing so, I will describe the compactified Jacobian of a curve with axis-like singularities, a result that is of independent interest.

• F. Reede: "Line bundles on noncommutative surfaces"

ABSTRACT: In this talk we will shortly describe the concept behind noncommutative surfaces. We are interested in line bundles on these surfaces. There is a moduli space classifying such line bundles, which can be seen as a generalization of the usual Picard scheme. In examples we will study these moduli spaces and see that there is a lot of hidden classic geometry surrounding these surfaces and moduli spaces.

• S. Rollenske: "Pluricanonical maps of stable surfaces"

ABSTRACT: In perfect analogy to the case of curves, stable surfaces can be defined as the class of surfaces needed for a modular compactification of the moduli space of surfaces of general type (with canonical singularities).

I will report on joint work with Wenfei Liu, where we study the pluricanonical maps of such stable surfaces, generalising the classical result of Bombieri. If time permits I will also mention some other botanical or geographical questions related to stable surfaces.

• P. Sosna: "On the Jordan-Hölder property for geometric derived categories"

ABSTRACT: We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan-Hölder property. More precisely, we will show that there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. This is joint work with C. Böhning und H.-C. Graf von Bothmer.

Registration

To register, please send an email mentioning your name, affiliation and whether you want to attend the conference dinner to Mrs Metzler.

Fee

There will be no conference fee.

Hotels

There is a limited capacity at Seminaris CampusHotel and Best Western Steglitz. Please make your own reservation. Details will be send to you with registration.

Other possibilities include Hotel Am Wilden Eber and Metropolitan Berlin.

Travel Information

How to get to the institute.

How to get to Zuse-Institut Berlin.

Participants

• Ane Anema (Groningen)
• A. Apostolov (Hannover)
• Nurömür Hülya Argüz (Hamburg)
• Christian Böhning (Hamburg)
• Hans-Christian v. Bothmer (Hamburg)
• Nathan Broomhead (Hannover)
• Chiara Camere (Hannover)
• Lennart Claus (FU Berlin)
• Wolfgang Ebeling (Hannover)
• Andreas Hochenegger (Köln)
• Klaus Hulek (Hannover)
• Sotiris Karanikolopoulos (FU Berlin)
• Lars Kastner (FU Berlin)
• Stefan Keil (HU Berlin)
• Remke Kloosterman (HU Berlin)
• Sebastian Krug (Hamburg)
• Niels Lindner (HU Berlin)
• Michael Lönne (Hannover)
• Elena Martinengo (FU Berlin)
• Fabian Müller (HU Berlin)
• Nicola Pagani (Hannover)
• Stefano Pascolutti (Hannover)
• David Ploog (Hannover)
• Juan Pons Llopis (FU Berlin)
• Fabian Reede (Göttingen)
• Sönke Rollenske (Bielefeld )
• Matthias Schütt (Hannover)
• Bernd Siebert (Hamburg)
• Pawel Sosna (Hamburg)
• Nicola Tarasca (Hannover)
• Matteo Tommasini (Hannover)
• Jaap Top (Groningen)
• Hung Ming Tsoi (Hamburg)
• Benjamin Wieneck (Hannover)