Summer School on Linear Systems (March 2009)

This is a summer school of the SFB/TR 45 Bonn-Essen-Mainz financed by Deutsche Forschungsgemeinschaft. It takes place March 23-27, 2009 at the Pedagogical University of Cracow. The workshop intends to improve the training of PhD students and postdocs in the area, in particular of the members of the SFB/TR 45.

Speakers

• Brian Harbourne, University of Nebraska (Lincoln, USA)
• Jun-Muk Hwang, Korea Institute for Advanced Study (Korea)
• Joaquim Roe, Universitat Autonoma de Barcelona (Spain)

Program

Hwang will discuss the conjecture on the degrees of Fano manifolds of Picard number 1. Two different approaches to this problem, due to Alan Nadel and Ziv Ran respectively, will be explained. Comparison with Fujita conjecture and Ein-Lazarsfeld conjecture on maximal Seshadri number will be given.

Harbourne and Roe will talk about relations between special linear systems and some topics in commutative algebra. Both lecture series are linked via multiplier ideals.   Harbourne's lectures are motivated by a number of long-standing open problems related to global aspects of the geometry of surfaces. The most fundamental of these is whether for each smooth projective surface there is a lower bound for the set of self-intersections of reduced curves on the surface. Another problem is related to a well known conjecture of Nagata. Studying these problems leads naturally to certain asymptotic invariants of Chudnovsky and Demailly, which can in turn be applied to problems in commutative algebra and algebraic geometry which have been studied using multiplier ideals and tight closure. Harbourne's lectures will discuss these invariants and some of their applications.           

Roé's lectures will cover specialization and degeneration techniques that have proved useful in the study of linear systems with base points in general position, and hence for the understanding of long-standing open problems related to the geometry of surfaces. The focus will be on blow-up techniques, including some classical results on complete ideals (local study) and their application to the global geometry, parameter spaces for blowups, and degenerations to reducible surfaces.
                                                                                                    

Prerequisites

We will assume the participants have some knowledge of algebraic geometry (Hartshorne) and have had a look at the chapters 2, 5, 9, and 10 of Lazarsfeld's "Positivity in Algebraic Geometry I + II" (Springer).

Schedule here

The registration has been closed ! 

Full financial support is only available for members of the SFB/TR 45.

Accomodation

All participants will be staying at the Student Hostel "Za Kolumnami", which is near the city center (5 tram stops from Rynek (main market square), see the map). Please indicate in the application form if you want us to reserve a room for you.

Travel Information

All lectures will take place at the Pedagogical University of Cracow.

Arriving by plane

The nearest airport is located at Kraków-Balice. Find your way from the airport to the city center here.

There is another airport, Katowice-Pyrzowice, from where the bus trip to Cracow lasts about two hours. (It is recommended to buy the bus tickets in advance, for example here.)

Arriving by train

The train station is Kraków Główny. Usually, train connections from Germany go via Berlin or Vienna.

Further information

For further questions, please contact the organizers:

• Tomasz Szemberg (, for local questions about Cracow, the accomodation or the program)
  
• Stefan Müller-Stach (, for global questions such as funding),

or just put your questions in the "further comments"-field in the registration form.