Finite Subgroups of the Plane Cremona Group
Igor V. Dolgachev
1
and Vasily A.
1
Department of Mathematics, University of Michigan, 525 E. University Ave., Ann
Arbor, MI, 49109
idolga@umich.edu
To Yuri I. Manin
Summary.
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.
Key words:
Cremona group, Del Pezzo surfaces, conic bundles.
2000 Mathematics Subject Classifcations
: 14E07 (Primary); 14J26, 14J50,
20B25 (Secondary)
1 Introduction
The Cremona group Cr
k
(
n
)
over a Feld
k
is the group of birational au-
tomorphisms of the projective space
P
n
k
, or equivalently, the group of
k
-automorphisms of the Feld
k
(
x
1
,x
2
,...,x
n
)
of rational functions in
n
independent variables. The group Cr
k
(1)
is the group of automorphisms of
the projective line, and hence it is isomorphic to the projective linear group
PGL
k
(2)
. Already in the case
n
=2
the group Cr
k
(2)
is not well understood
in spite of extensive classical literature (e.g., [
21
], [
35
]) on the subject as well
as some modern research and expositions of classical results (e.g., [
2
]). Very
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
(2)
. We return to the classical problem of classi-
Fcation of Fnite subgroups of Cr
(2)
. The classiFcation of Fnite subgroups of
PGL
(2)
is well known and goes back to ±. Klein. It consists of cyclic, dihedral,
1
The author was supported in part by NS± grant 0245203.
2
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
,
443
Progress in Mathematics 269, DOI 10.1007/978-0-8176-4745-2_11,
c
±
Springer Science+Business Media, LLC 2009
Iskovskikh
2