Finite Subgroups of the Plane Cremona Group
Igor V. Dolgachev
and Vasily A.
Department of Mathematics, University of Michigan, 525 E. University Ave., Ann
Arbor, MI, 49109
To Yuri I. Manin
This paper completes the classic and modern results on classiFcation
of conjugacy classes of Fnite subgroups of the group of birational automorphisms of
the complex projective plane.
Cremona group, Del Pezzo surfaces, conic bundles.
2000 Mathematics Subject Classifcations
: 14E07 (Primary); 14J26, 14J50,
The Cremona group Cr
over a Feld
is the group of birational au-
tomorphisms of the projective space
, or equivalently, the group of
-automorphisms of the Feld
of rational functions in
independent variables. The group Cr
is the group of automorphisms of
the projective line, and hence it is isomorphic to the projective linear group
. Already in the case
the group Cr
is not well understood
in spite of extensive classical literature (e.g., [
]) on the subject as well
as some modern research and expositions of classical results (e.g., [
little is known about the Cremona groups in higher-dimensional spaces.
In this paper we consider the plane Cremona group over the Feld of com-
plex numbers, denoted by Cr
. We return to the classical problem of classi-
Fcation of Fnite subgroups of Cr
. The classiFcation of Fnite subgroups of
is well known and goes back to ±. Klein. It consists of cyclic, dihedral,
The author was supported in part by NS± grant 0245203.
The author was supported in part by R±BR 05-01-00353-a R±BR 08-01-00395-a,
grant CRD± RUMI 2692-MO-05 and grant of NSh 1987-2008.1.
Y. Tschinkel and Y. Zarhin (eds.),
Algebra, Arithmetic, and Geometry
Progress in Mathematics 269, DOI 10.1007/978-0-8176-4745-2_11,
Springer Science+Business Media, LLC 2009