39th Autumn School in Algebraic Geometry
COMPLEX AFFINE GEOMETRY
Łukęcin, POLAND, 18-24.09.2016
Takashi Kishimoto (Saitama University)
A. Dubouloz (Université de Bourgogne)
Karol Palka (IMPAN, Warsaw)
Organizers: M. Koras (University of Warsaw), K. Palka, T. Pełka (IMPAN, Warsaw).
Inquires should be directed to T. Pełka: tpelka (at) impan.pl.
The lectures will take place from Monday 19.09 to Friday 23.09. Each day there will be three one-hour lectures followed by an excercise session. The lectues will be aimed for graduate students and young researchers in algebraic geometry. A detailed program of the meeting will be posted later. The courses will be held in English.
Outline of the courses
1. Logarithmic Kodaira dimension and uniruledness (T. Kishimoto). The introductory part of the course will cover the notion of logartihmic Iitaka-Kodaira dimension, its basic properties and classical applications, such as finiteness of the automorphism groups of varieties of log general type. We will discuss the relation with affine ruledness and log-uniruledness, especially in lower dimensions and with the exsitence of a cyllindrical open subset. We will also study affine-ruled surfaces and their deformations.
2. The Cancellation Problem (A. Dubouloz). We will focus on the various approaches to the Cancellation Problem, which asks if a variety whose cylinder is an affine space is necessarily an affine space itself. We will explain how to use the results developed in the introductory part to give an affirmative answer in dimension two. Next, we will give an overwiew of counterexamples to the Generalized Cancellation Problem, starting from classical strategies by Danielewski and Hochster. We will discus the recent progress in the study of cancellation for higher-dimensional varieties. We will present some interesting constructions of exotic structures based on the well known candidates for counterexamples to cancellation like the Koras-Russell threefolds.
3. Recent progress in the geometry of Q-acyclic surfaces (K. Palka). We will review classical and new tools of the geometry of quasi-projective surfaces and log surfaces focusing on their applications to problems concerning Q-acyclic surfaces, including the recent solution of the Coolidge-Nagata conjecture and the progress in classification of planar rational cuspidal curves. The key role will be played by the notion of almost minimality (for integral and rational boundary divisors), which allows one to apply the techniques of the log Miniamal Model Program to log smooth pairs without losing log smoothness. We will also discuss very recent results on the equivariant Jacobian Conjecture for Q-acyclic surfaces.
Prerequisites: Basic knowledge of algebraic geometry.
- S. Iitaka, Algebraic geometry, GTM 76, Springer-Verlag, New York, 1982,
- M. Miyanishi, Open algebraic surfaces, CRM Monograph Series, vol. 12, AMS, Providence, RI, 2001.
- Articles on the above topics by the speakers.
The school will take place in a Warsaw University pension in Łukęcin (look here for more information), on the Western part of the Polish Baltic sea shore. Participants are expected to arrive on Sunday, September 18th, evening. Lectures are from Monday to Friday and Saturday is the departure day.
The accommodation (full board, double room) costs about 125 złoty (PLN) a day (1 Euro is approximately 4.4 złoty). Participants, especially graduate students and young researchers are encouraged to apply for accommodation cost waiver. Please indicate your need for support in the registration form. You will need to provide the name of your scientific advisor and we may ask you to provide an additional letter of recommendation.
A small registration fee of around 60 złoty per person will apply. The fee is not covered by the financial support.
Place and directions: nice pension near the Baltic see, good food, beach nearby, should be warm.
Here is a MAP OF ŁUKĘCIN with travel details
The nearest airport is Szczecin-Goleniów (Poland), you may consider also traveling through Berlin (Germany) and then taking a train to Szczecin (it takes about 2 hrs), see the German railway page. Or you may take a bus, see buses from Berlin airports to Szczecin and Koszalin.
The organizers will provide a conference bus from Szczecin Główny train station to Łukęcin, which will depart on Sunday, September 18th, around 19:30, and a bus back from Łukęcin to Szczecin on Saturday, September 24th, which will arrive to Sczecin Główny station before 11am.
Other possibilities to get from Szczecin to Łukecin are: a minibus, see e.g. KSKBus, or Ober-Trans, or a taxi. There is also a small airport in Heringsdorf, near Świnoujście, see here. It is located about 65 km from Łukęcin and it has a few flights from Germany, Austria, Switzerland and Poland. We have no information about arriving to and from this airport.
Previous schools: http://agvs.mimuw.edu.pl/schools/