Automorphisms in Birational and Affine Geometry: Levico Terme, Italy, October 2012

509 Pages·2014·5.34 MB·English

by  Ivan Cheltsov

Springer Proceedings in Mathematics & Statistics
Ivan Cheltsov
Ciro Ciliberto
Hubert Flenner
James McKernan
Yuri G. Prokhorov
Mikhail Zaidenberg E ditors
Automorphisms
in Birational and
Affi ne Geometry
Levico Terme, Italy, October 2012
Springer Proceedings in Mathematics & Statistics
Volume 79
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Ivan Cheltsov • Ciro Ciliberto • Hubert Flenner •
James McKernan • Yuri G. Prokhorov •
Mikhail Zaidenberg
Editors
Automorphisms in Birational
and Affine Geometry
Levico Terme, Italy, October 2012
123
Editors
IvanCheltsov CiroCiliberto
SchoolofMathematics DepartmentofMathematics
UniversityofEdinburgh UniversityofRomeTorVergata
Edinburgh,UnitedKingdom Rome,Italy
HubertFlenner JamesMcKernan
FacultyofMathematics DepartmentofMathematics
RuhrUniversityBochum UniversityofCaliforniaSanDiego
Bochum,Germany LaJolla,California,USA
YuriG.Prokhorov MikhailZaidenberg
SteklovMathematicalInstitute InstitutFourierdeMathématiques
RussianAcademyofSciences UniversitéGrenobleI
Moscow,Russia Grenoble,France
ISSN2194-1009 ISSN2194-1017(electronic)
ISBN978-3-319-05680-7 ISBN978-3-319-05681-4(eBook)
DOI10.1007/978-3-319-05681-4
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014941702
MathematicsSubjectClassification(2010):14L30,14E07,14R20,14R10,32M17
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Preface
The conference brought together specialists from the birational geometry of pro-
jectivevarieties,affinealgebraicgeometry,andcomplexalgebraicgeometry.Topics
fromtheseareasincludeMoritheory,Cremonagroups,algebraicgroupactions,and
automorphisms. The ensuing talks and the discussions have highlighted the close
connections between these areas. The meeting allowed these groups to exchange
knowledgeandtolearnmethodsfromadjacentfields.Fordetailedinformationabout
thisconference,seehttp://www.science.unitn.it/cirm/GABAG2012.html.
The chapters in this book cover a wide area of topics from classical algebraic
geometry to birational geometry and affine geometry, with an emphasis on group
actionsandautomorphismgroups.
Amongthetotalof27chapters,eightgiveanoverviewofanareainoneofthe
fields. The otherscontainoriginalcontributions,or mix originalcontributionsand
surveys.
In the birational part, there are chapters on Fano and del Pezzo fibrations,
birationalrigidityandsuperrigidityofFanovarieties,birationalmorphismsbetween
threefolds,subgroupsof the Cremona group, Jordan groups, real Cremona group,
andarealversionoftheSarkisovprogram.
In the part on classical projective geometry, there are chapters devoted to
algebraic groups acting on projective varieties, automorphism groups of moduli
spaces, and the algebraicity problem for analytic compactifications of the affine
plane.
The topics in affine geometry include: different aspects of the automorphism
groups of affine varieties, flexibility properties in affine algebraic and analytic
geometry,automorphismsof affinespaces andShestakov–Umirbaevtheory,affine
geometry in positive characteristic, the automorphism groups of configuration
spacesandotherclassesofaffinevarietiesanddeformationsofcertaingroupactions
onaffinevarieties.
Hopefully these proceedingswill be of help for both specialists, who will find
information on the current state of research, and young researchers, who wish to
learnaboutthesefascinatingandactiveareasofmathematics.
v
vi Preface
Acknowledgments
The editors would like to thank the contributorsof this volume.They are grateful
to the CIRM at Trento, its directors Marco Andreatta and Fabrizio Catanese, its
secretary Augusto Micheletti, and the whole staff for their permanent support
in the organization of the conference and the distribution of this volume. Our
thanks are due also to Marina Reizakis, Frank Holzwarth, and the staff of the
Springerpublishinghouse for agreeablecooperationduringthe preparationof the
proceedings.
Edinburgh,UK IvanCheltsov
Rome,Italy CiroCiliberto
Bochum,Germany HubertFlenner
LaJolla,CA JamesMcKernan
Moscow,Russia YuriG.Prokhorov
Grenoble,France MikhailZaidenberg
Contents
PartI BirationalAutomorphisms
SingulardelPezzoFibrationsandBirationalRigidity....................... 3
HamidAhmadinezhad
AdditiveActionsonProjectiveHypersurfaces................................ 17
IvanArzhantsevandAndreyPopovskiy
CremonaGroupsofRealSurfaces............................................. 35
JérémyBlancandFrédéricMangolte
OnAutomorphismsandEndomorphismsofProjectiveVarieties.......... 59
MichelBrion
DelPezzoSurfacesandLocalInequalities .................................... 83
IvanCheltsov
FanoHypersurfacesandtheirBirationalGeometry......................... 103
TommasodeFernex
OntheLocusofNonrigidHypersurfaces ..................................... 121
ThomasEcklandAleksandrPukhlikov
OntheGenusofBirationalMapsBetweenThreefolds...................... 141
StéphaneLamy
OntheAutomorphismsofModuliSpacesofCurves......................... 149
AlexMassarentiandMassimilianoMella
NormalAnalyticCompactificationsofC2..................................... 169
PinakiMondal
JordanGroupsandAutomorphismGroupsofAlgebraicVarieties........ 185
VladimirL.Popov
vii
viii Contents
2-ElementarySubgroupsoftheSpaceCremonaGroup..................... 215
YuriProkhorov
BirationalAutomorphismGroupsof ProjectiveVarieties
ofPicardNumberTwo .......................................................... 231
De-QiZhang
PartII AutomorphismsofAffineVarieties
RationalCurveswithOnePlaceatInfinity ................................... 241
AbdallahAssi
The Jacobian Conjecture, Together with Specht
andBurnside-TypeProblems................................................... 249
AlexeiBelov,LeonidBokut,LouisRowen,andJie-TaiYu
Equivariant Triviality of Quasi-Monomial Triangular
G -ActionsonA4................................................................. 287
a
AdrienDubouloz,DavidR.Finston,andImadJaradat
AutomorphismGroupsofCertainRationalHypersurfaces
inComplexFour-Space.......................................................... 301
AdrienDubouloz,LucyMoser-Jauslin,andPierre-MariePoloni
LaurentCancellationforRingsofTranscendenceDegreeOne
OveraField....................................................................... 313
GeneFreudenburg
DeformationsofA1-Fibrations ................................................. 327
RajendraV.Gurjar,KayoMasuda,andMasayoshiMiyanishi
RemarkonDeformationsofAffineSurfaceswithA1-Fibrations........... 363
TakashiKishimoto
HowtoProvetheWildnessofPolynomialAutomorphisms:
AnExample....................................................................... 381
ShigeruKuroda
Flexibility Properties in Complex Analysis and Affine
AlgebraicGeometry ............................................................. 387
FrankKutzschebauch
StronglyResidualCoordinatesoverAŒx(cid:2) ..................................... 407
DrewLewis
Configuration Spaces of the Affine Line and their
AutomorphismGroups.......................................................... 431
VladimirLinandMikhailZaidenberg
Contents ix
OntheNewtonPolygonofaJacobianMate .................................. 469
LeonidMakar-Limanov
KellerMapsofLowDegreeoverFiniteFields................................ 477
StefanMaubachandRoelWillems
Cancellation....................................................................... 495
PeterRussell