Cremona Groups and the Icosahedron

521 Pages·2015·2.611 MB·English

by  Ivan Cheltsov, Constantin Shramov

Mathematics
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
C
Cremona Groups and the Icosahedron focuses on the Cre- r
e
mona groups of ranks 2 and 3 and describes the beautiful m Cremona Groups
appearances of the icosahedral group 𝔄 in them. The book o
5 n
surveys known facts about surfaces with an action of 𝔄 , ex- a
5 and the Icosahedron
plores 𝔄 -equivariant geometry of the quintic del Pezzo three- G
5
fold V , and gives a proof of its 𝔄 -birational rigidity. r
5 5 o
u
The authors explicitly describe many interesting 𝔄 -invariant p
5
s
subvarieties of V , including 𝔄 -orbits, low-degree curves,
5 5 a
invariant anticanonical K3 surfaces, and a mildly singular sur- n
d
face of general type that is a degree five cover of the diago-

t
nal Clebsch cubic surface. They also present two birational h
e
selfmaps of V that commute with 𝔄 -action and use them to
5 5 I
c
determine the whole group of 𝔄 -birational automorphisms.
o
5
As a result of this study, they produce three non-conjugate s
a
h
icosahedral subgroups in the Cremona group of rank 3, one
e
of them arising from the threefold V . d
5 r
o
This book presents up-to-date tools for studying birational n
geometry of higher-dimensional varieties. In particular, it pro-
vides you with a deep understanding of the biregular and
birational geometry of V .
5
SC
Ivan Cheltsov
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s Constantin Shramov
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K23832
www.crcpress.com
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MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Cremona Groups
and the Icosahedron
Ivan Cheltsov
University of Edinburgh
Scotland, UK
Constantin Shramov
Steklov Mathematical Institute and Higher School of Economics
Moscow, Russia
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MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Series Editors
John A. Burns
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Contents
Preface xi
Acknowledgments xv
Notation and conventions xvii
1 Introduction 1
1.1 Conjugacy in Cremona groups. . . . . . . . . . . . . . . . . 1
1.2 Three-dimensional projective space . . . . . . . . . . . . . . 5
1.3 Other rational Fano threefolds . . . . . . . . . . . . . . . . 8
1.4 Statement of the main result . . . . . . . . . . . . . . . . . 11
1.5 Outline of the book . . . . . . . . . . . . . . . . . . . . . . . 12
I Preliminaries 21
2 Singularities of pairs 23
2.1 Canonical and log canonical singularities . . . . . . . . . . . 23
2.2 Log pairs with mobile boundaries . . . . . . . . . . . . . . . 26
2.3 Multiplier ideal sheaves . . . . . . . . . . . . . . . . . . . . 28
2.4 Centers of log canonical singularities . . . . . . . . . . . . . 30
2.5 Corti’s inequality . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Noether–Fano inequalities 39
3.1 Birational rigidity . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Fano varieties and elliptic fibrations . . . . . . . . . . . . . 40
3.3 Applications to birational rigidity . . . . . . . . . . . . . . . 46
3.4 Halphen pencils . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
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vi CONTENTS
4 Auxiliary results 53
4.1 Zero-dimensional subschemes . . . . . . . . . . . . . . . . . 53
4.2 Atiyah flops . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 One-dimensional linear systems . . . . . . . . . . . . . . . . 59
4.4 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
II Icosahedral group 65
5 Basic properties 67
5.1 Action on points and curves . . . . . . . . . . . . . . . . . . 67
5.2 Representation theory . . . . . . . . . . . . . . . . . . . . . 71
5.3 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Curves of low genera . . . . . . . . . . . . . . . . . . . . . . 77
5.5 SL (C) and PSL (C) . . . . . . . . . . . . . . . . . . . . . . 79
2 2
5.6 Binary icosahedral group. . . . . . . . . . . . . . . . . . . . 87
5.7 Symmetric group . . . . . . . . . . . . . . . . . . . . . . . . 89
5.8 Dihedral group . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Surfaces with icosahedral symmetry 95
6.1 Projective plane . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Quintic del Pezzo surface . . . . . . . . . . . . . . . . . . . 105
6.3 Clebsch cubic surface . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Two-dimensional quadric . . . . . . . . . . . . . . . . . . . 123
6.5 Hirzebruch surfaces . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 Icosahedral subgroups of Cr (C) . . . . . . . . . . . . . . . 140
2
6.7 K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
III Quintic del Pezzo threefold 151
7 Quintic del Pezzo threefold 153
7.1 Construction and basic properties . . . . . . . . . . . . . . . 153
7.2 PSL (C)-invariant anticanonical surface . . . . . . . . . . . 159
2
7.3 Small orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.5 Orbit of length five . . . . . . . . . . . . . . . . . . . . . . . 174
7.6 Five hyperplane sections . . . . . . . . . . . . . . . . . . . . 178
7.7 Projection from a line . . . . . . . . . . . . . . . . . . . . . 183
7.8 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
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CONTENTS vii
8 Anticanonical linear system 189
8.1 Invariant anticanonical surfaces . . . . . . . . . . . . . . . . 189
8.2 Singularities of invariant anticanonical surfaces . . . . . . . 198
8.3 Curves in invariant anticanonical surfaces . . . . . . . . . . 203
9 Combinatorics of lines and conics 207
9.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.2 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10 Special invariant curves 221
10.1 Irreducible curves . . . . . . . . . . . . . . . . . . . . . . . . 221
10.2 Preliminary classification of low degree curves . . . . . . . . 230
11 Two Sarkisov links 239
11.1 Anticanonical divisors through the curve L . . . . . . . . . 239
6
11.2 Rational map to P4 . . . . . . . . . . . . . . . . . . . . . . . 244
11.3 A remarkable sextic curve . . . . . . . . . . . . . . . . . . . 255
11.4 Two Sarkisov links . . . . . . . . . . . . . . . . . . . . . . . 260
11.5 Action on the Picard group . . . . . . . . . . . . . . . . . . 269
IV Invariant subvarieties 279
12 Invariant cubic hypersurface 281
12.1 Linear system of cubics . . . . . . . . . . . . . . . . . . . . 281
12.2 Curves in the invariant cubic . . . . . . . . . . . . . . . . . 282
12.3 Bring’s curve in the invariant cubic . . . . . . . . . . . . . . 288
12.4 Intersecting invariant quadrics and cubic . . . . . . . . . . . 293
12.5 A remarkable rational surface . . . . . . . . . . . . . . . . . 306
13 Curves of low degree 313
13.1 Curves of degree 16 . . . . . . . . . . . . . . . . . . . . . . . 313
13.2 Six twisted cubics. . . . . . . . . . . . . . . . . . . . . . . . 320
13.3 Irreducible curves of degree 18 . . . . . . . . . . . . . . . . 326
13.4 A singular curve of degree 18 . . . . . . . . . . . . . . . . . 332
13.5 Bring’s curve . . . . . . . . . . . . . . . . . . . . . . . . . . 337
13.6 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 339
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viii CONTENTS
14 Orbits of small length 347
14.1 Orbits of length 20 . . . . . . . . . . . . . . . . . . . . . . . 347
14.2 Ten conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
14.3 Orbits of length 30 . . . . . . . . . . . . . . . . . . . . . . . 363
14.4 Fifteen twisted cubics . . . . . . . . . . . . . . . . . . . . . 370
15 Further properties of the invariant cubic 375
15.1 Intersections with low degree curves . . . . . . . . . . . . . 375
15.2 Singularities of the invariant cubic . . . . . . . . . . . . . . 377
15.3 Projection to Clebsch cubic surface . . . . . . . . . . . . . . 388
15.4 Picard group . . . . . . . . . . . . . . . . . . . . . . . . . . 397
16 Summary of orbits, curves, and surfaces 405
16.1 Orbits vs. curves . . . . . . . . . . . . . . . . . . . . . . . . 405
16.2 Orbits vs. surfaces . . . . . . . . . . . . . . . . . . . . . . . 406
16.3 Curves vs. surfaces . . . . . . . . . . . . . . . . . . . . . . . 406
16.4 Curves vs. curves . . . . . . . . . . . . . . . . . . . . . . . . 410
V Singularities of linear systems 413
17 Base loci of invariant linear systems 415
17.1 Orbits of length 10 . . . . . . . . . . . . . . . . . . . . . . . 415
17.2 Linear system Q . . . . . . . . . . . . . . . . . . . . . . . . 416
3
17.3 Isolation of orbits in S . . . . . . . . . . . . . . . . . . . . 423
17.4 Isolation of arbitrary orbits . . . . . . . . . . . . . . . . . . 428
17.5 Isolation of the curve L . . . . . . . . . . . . . . . . . . . 434
15
18 Proof of the main result 443
18.1 Singularities of linear systems . . . . . . . . . . . . . . . . . 443
18.2 Restricting divisors to invariant quadrics . . . . . . . . . . . 444
18.3 Exclusion of points and curves different from L . . . . . . 451
15
18.4 Exclusion of the curve L . . . . . . . . . . . . . . . . . . . 453
15
18.5 Alternative approach to exclusion of points . . . . . . . . . 456
18.6 Alternative approach to the exclusion of L . . . . . . . . . 461
15
19 Halphen pencils and elliptic fibrations 465
19.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . 465
19.2 Exclusion of points . . . . . . . . . . . . . . . . . . . . . . . 469
19.3 Exclusion of curves . . . . . . . . . . . . . . . . . . . . . . . 471
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