35th Autumn School in Algebraic Geometry
Subgroups of Cremona groups
Lukecin, Poland, September 23 -- September 29, 2012
Teachers: Jeremy Blanc (Basel) and Yuri Prokhorov (Moscow)
Abstract: The subject of the school concerns finite subgroups of
groups of birational transformations of projective spaces. This topic is
classical and goes back to the last quarter of the 19th century and it
has its vivid renaissance in the last two decades. Two series of
lectures will survey the most recent methods and results in this
field. The lectures of Yuri Prokhorov will focus on Fano manifolds and
will provide an introduction to Mori theory. The lectures of Jeremy
Blanc will concentrate on finite subgroups of birational transformations
of the plane.
Prerequisites: Basic knowledge of algebraic geometry.
Program of the school: There will be 2 lectures each morning,
90 min each, followed by 90 min excercise session in the afternoon and
contributed talks in the evening.
Fano varieties and Cremona groups, by Yuri Prokhorov.
Abstract (subject to change): The aim of this course is to present
basic results on Fano varieties and give examples; it will also serve
as a preparation for the course of Jeremy Blanc. The following topics
will be covered:
Readings to Prokhorov's lectures:
- Basic properties of Fano varieties. Examples. Del Pezzo
- Introduction to Mori theory, Fano varieties in the framework of
- Three-dimensional MMP. Outline of Mori-Mukai classification.
- Introduction to Sarkisov links. Examples. Outline of Iskovskikh
classification (three-dimensional case).
- Application to the classification of finite subgroups of Cremona
- Fano varieties:
- Iskovskikh, Prokhorov, Fano varieties. Algebraic
geometry, V, 1-247, Encyclopaedia Math. Sci., 47, Springer, Berlin,
- Mukai, New developments in the theory of Fano threefolds:
vector bundle method and moduli problems Sugaku Expositions 15
(2002), no. 2, 125-150, available
- Del Pezzo surfaces:
- Manin, Cubic forms: algebra, geometry, arithmetic. North-Holland
Mathematical Library, Vol. 4. 1974 [Ch. 4, sect. 23-26]
- Smith, Karen E.; Corti, Alessio Rational and nearly rational
varieties. Cambridge Studies in Advanced Mathematics, 92. Cambridge
University Press, Cambridge, 2004
- Dolgachev, Classical algebraic geometry: a modern view,
http://www.math.lsa.umich.edu/~idolga/, [Ch. 8]
Finite subgroups of the Cremona group in dimension 2, by Jeremy Blanc.
Abstract: This course will be devoted to the study of finite
subgroups of the Cremona group in dimension 2. This is closely related
to the G-equivariant MMP in dimension 2, which has the advantage to be
really easier than in dimensions higher (and in fact proved far before
The course will describe the geometry on del Pezzo surfaces and conic
bundles with a group action, and relation between all these models. If
time permits, the case of non-algebraically closed field will also be
Readings to lectures by Blanc.
Organizers: Jaroslaw Buczynski, Jaroslaw Wisniewski, Institute
of Mathematics, Warsaw University.
- Blanc, Finite subgroups
of the Cremona group of the plane. Notes to the course
- Dolgachev, Finite subgroups of the plane Cremona group
Algebraic geometry in East Asia, Seoul 2008, 1-49, Adv. Stud. Pure
Math., 60, Math. Soc. Japan, Tokyo,
- Dolgachev, Iskovskikh, Finite subgroups of the plane Cremona
group. Algebra, arithmetic, and geometry: in honor of
Yu. I. Manin. Vol. I, 443-548, Progr. Math., 269, Birkhauser Boston,
Inc., Boston, MA,
- Bayle, Beauville, Birational involutions of P2. Kodaira's
issue. Asian J. Math. 4 (2000), no. 1, 1-117.
- de Fernex, On planar Cremona maps of prime order. Nagoya
Math. J. 174 (2004), 1-28.
- Blanc, Elements and cyclic subgroups of finite order of the
Cremona group. Comment. Math. Helv. 86 (2011), no. 2, 469-497.
- Serre, Le groupe de Cremona et ses sous-groupes
finis Seminaire Bourbaki. Volume 2008/2009. Exposes 997-1011.
The joint picture of participants