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
[email protected]
To Yuri I. Manin
Summary.
This paper completes the classic and modern results on classification
of conjugacy classes of finite subgroups of the group of birational automorphisms of
the complex projective plane.
Key words:
Cremona group, Del Pezzo surfaces, conic bundles.
2000 Mathematics Subject Classifications
: 14E07 (Primary); 14J26, 14J50,
20B25 (Secondary)
1 Introduction
The Cremona group Cr
k
(
n
)
over a field
k
is the group of birational au
tomorphisms
of
the
projective
space
P
n
k
,
or
equivalently,
the
group
of
k
automorphisms of the field
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 higherdimensional spaces.
In this paper we consider the plane Cremona group over the field of com
plex numbers, denoted by Cr
(2)
. We return to the classical problem of classi
fication of finite subgroups of Cr
(2)
. The classification of finite subgroups of
PGL
(2)
is well known and goes back to F. Klein. It consists of cyclic, dihedral,
1
The author was supported in part by NSF grant 0245203.
2
The author was supported in part by RFBR 050100353a RFBR 080100395a,
grant CRDF RUMI 2692MO05 and grant of NSh 19872008.1.
Y. Tschinkel and Y. Zarhin (eds.),
Algebra, Arithmetic, and Geometry
,
443
Progress in Mathematics 269, DOI 10.1007/9780817647452_11,
c Springer Science+Business Media, LLC 2009
Iskovskikh
2
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444
Igor V. Dolgachev and Vasily A. Iskovskikh
tetrahedral, octahedral, and icosahedral groups. Groups of the same type and
order constitute a unique conjugacy class in PGL
(2)
. Our goal is to find a
similar classification in the twodimensional case.
The history of this problem begins with the work of E. Bertini [
10
],
who classified conjugacy classes of subgroups of order 2 in Cr
(2)
. Already
in this case the answer is drastically different. The set of conjugacy classes
is parametrized by a disconnected algebraic variety whose connected compo
nents are respectively isomorphic to either the moduli spaces of hyperelliptic
curves of genus
g
(de Jonquières involutions), or the moduli space of canonical
curves of genus 3 (Geiser involutions), or the moduli space of canonical curves
of genus 4 with vanishing theta characteristic (Bertini involutions). Bertini’s
proof was considered to be incomplete even according to the standards of
rigor of nineteenthcentury algebraic geometry. A complete and short proof
was published only a few years ago by L. Bayle and A. Beauville [
5
].
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 Fall '04
 IgorDolgachev
 Math, Group Theory, Birational geometry, Plane Cremona Group

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