Contributions to Algebraic Geometry: Impanga Lecture Notes

516 Pages·2012·9.822 MB·English

by  Piotr Pragacz

EMS Series of Congress Reports
EMS Congress Reportspublishes volumes originating from conferences or seminars focusing on any field
of pure or applied mathematics.The individual volumes include an introduction into their subject and
review of the contributions in this context.Articles are required to undergo a refereeing process and are
accepted only if they contain a survey or significant results not published elsewhere in the literature.
Previously published:
Trends in Representation Theory of Algebras and Related Topics,Andrzej Skowron´ski (ed.)
K-Theory and Noncommutative Geometry,Guillermo Cortiñaset al. (eds.)
Classification of Algebraic Varieties,Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.)
Surveys in Stochastic Processes,Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.)
Representations of Algebras and Related Topics,Andrzej Skowron´ski and Kunio Yamagata (eds.)
Contributions to
Algebraic Geometry
Impanga Lecture Notes
Piotr Pragacz
Editor
Editor:
Piotr Pragacz
Institute of Mathematics
Polish Academy of Sciences
ul.S´niadeckich 8
00-956 Warszawa
Poland
E-mail:[email protected]
2010 Mathematics Subject Classification:11S15,13D10,14-02,14B05,14B12,14C17,14C20,14C35,
14D06,14D15,14D20,14D23,14E15,14E30,14F43,14H10,14H40,14H42,14J17,14J28,14J30,
14J32,14J50,14J70,14K10,14K25,14L30,14M15,14M17,14M25,14N10,14N15,32G10,32Q45,
34A30,53D05,55N91;01-02,01A70,05E05,11S85,13A35,13D10,14C30,14F18,14J26,14J32,
14N20,19D55,32S25,53D20,57R45
Key words:K3 surface,Enriques surface,Calabi–Yau threefold,linear system,Seshadri constant,multi-
plier ideal,differential form,log canonical treshold,Mori theory,canonical ring,deformation of mor-
phism,Hodge numbers,logarithmic differential form,Prym variety,moduli space,syzygy,Wron´ski
determinant,linear ODE,ramification locus,Schubert calculus,Grassmannian,Schubert variety,Schur
function,singularity,Thom polynomial,P-ideal,equivariant localization,toric variety,symplectic mani-
fold,toric stack,symplectic quotient,equivariant cohomology,Bloch group,field extension,norm
ISBN 978-3-03719-114-9
The Swiss National Library lists this publication in The Swiss Book,the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright.All rights are reserved,whether the whole or part of the material
is concerned,specifically the rights of translation,reprinting,re-use of illustrations,recitation,
broadcasting,reproduction on microfilms or in other ways,and storage in data banks.For any kind of
use permission of the copyright owner must be obtained.
© 2012 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
ETH-Zentrum SEW A27
CH-8092 Zürich
Switzerland
Phone:[phone]
Email:[email protected]
Homepage:www.ems-ph.org
Typeset using the authors’ TEX files:I.Zimmermann,Freiburg
Printing and binding:Beltz Bad Langensalza GmbH,Bad Langensalza,Germany
∞Printed on acid free paper
9 8 7 6 5 4 3 2 1
A tribute to Oscar Zariski

Preface
ThisbookisanoutgrowthoftheImpangaConferenceonAlgebraicGeometry,heldat
theBanachCenterinBe¸dlewoonJuly4–10,2010. Fordetailedinformationaboutthis
event,see
http://www.impan.pl/~impanga/school
The articles in this volume cover a broad range of topics in algebraic geometry:
classicalvarieties,linearsystems,birationalgeometry,MinimalModelProgram,mod-
ulispaces,toricvarieties,enumerativetheoryofsingularities,equivariantcohomology
andarithmeticquestions.
The book is based on contributions by conference speakers and participants, in-
cludingtwoarticlesbymathematicianswhowereunabletoattendthemeeting. Here
isabriefsummaryofthecontentofthevolume.
ThearticlebyKlausAltmannetal.describesthelanguageofpolyhedraldivisors,
whichisusedwhenworkingwithT-varieties. Thislanguageisexplainedinparallelto
thewellestablishedtheoryoftoricvarieties. Inadditiontobasicconstructions,subjects
discussedinclude: singularities, separatednessandproperness, divisorsandintersec-
tiontheory,cohomology,Coxrings,polarizationsandequivariantdeformations.
DaveAndersongivesanextensiveintroductiontoequivariantcohomologyinalge-
braicgeometry. Thefirstpartisanoverview,includingbasicdefinitionsandexamples.
In the second part, the author discusses one of the most useful aspects of the the-
ory: thepossibilityoflocalizingatfixedpointswithoutlosinginformation. Thethird
part focuses on Grassmannians, and describes some recent positivity results for their
equivariantcohomologyrings.
ThearticlebyThomasBaueretal.discussesopenproblemsinthetheoryoflinear
systems. Thediscussionrevolvesaroundthespecialityandthepostulationproblems
as well as the containment problems for various powers of ideals. The main moti-
vations come from the Harbourne–Hirschowitz, Nagata and the Bounded Negativity
Conjectures.
GergelyBércziinvestigatesthemodulispaceofholomorphicmapgermsfromthe
complexlineintocomplexcompactmanifolds. Twomajorapplicationsarediscussed:
to Thom polynomials for Morin singularities in global singularity theory, and to the
Green–Griffithsconjectureinthetheoryofhyperbolicalgebraicvarieties.
ThearticlebyPaoloCasciniandVladmirLazic´givesanintroductiontosomerecent
developments in Mori theory aimed for a less experienced reader. In particular, the
authors’recentdirectproofofthefinitegenerationofthecanonicalringisdiscussed.
SławomirCynkandSławomirRamspresentvariousresultsoninvariantsofareso-
lutionofasingularprojectivehypersurface. Inparticular,theyprovesomedefect-type
formulas. Theproofsuse,inanessentialway,thesheavesoflogarithmicdifferential
forms.
viii Preface
GavrilFarkasstudiesthegeometryofthemodulispaceofPrymvarieties. Several
applications of Prym varieties in algebraic geometry are presented. The exposition
begins with a historical discussion of the life and achievements of Friedrich Prym.
Topics treated in subsequent sections include singularities and Kodaira dimension of
the moduli space, syzygies of Prym-canonical embedding and the geometry of the
modulispaceR forsmallgenus.
g
ThearticlebyLetterioGattoandInnaScherbakconcernsWron´skiansinalgebraic
geometry. Wron´skians,usuallyintroducedinstandardcoursesinOrdinaryDifferential
Equations(ODE),areausefultoolinalgebraicgeometrytodetectramificationlociof
linear systems. The authors describe some “materializations” of theWron´skian, and
ofitscloserelatives,thegeneralizedWron´skians(labelledbypartitions),inalgebraic
geometry. EmphasisisputontherelationshipbetweenSchubertcalculusandODE.
ThearticlebyKevinHitchinsonandMashaVlasenkosupportsBurtTotaro’sexpec-
tationthatBloch’shigherChowgroupsofafieldkcanbecomputedusingaverysmall
classofaffinealgebraicvarieties(linearspacesintherightcoordinates),whereasthe
currentdefinitionusesessentiallyallalgebraiccyclesinaffinespace. Theauthorscon-
siderasimplemodificationofCH2.Spec.k/;3/usingonlylinearsubvarietiesinaffine
spaces, and show that it maps surjectively to the Bloch group B.k/ for any infinite
fieldk. Theyalsodescribethekernelofthismap.
AndreasHocheneggerandFrederikWittdescribesomeconstructionsinsymplectic
toric geometry. The starting point is the Delzant construction of a symplectic toric
manifoldfromasmoothpolytope. Thearticlethendiscussestherefinementsforrational
butnotnecessarilysmoothpolytopesduetoLermanandTolman,leadingtosymplectic
toricorbifoldsor,moregenerally,symplectictoricDMstacks. Theauthorsshowthat
thelatterstacksareisomorphictothestacksobtainedbyBorisovetal.wheneverthe
stackyfanisinducedbyapolytope.
ClemensJörderandStefanKebekusaddressthefollowingquestion. Letf W Y !
X betheinclusionmapofacompactreducedsubspaceofacomplexmanifold,andlet
F (cid:2) T be a subsheaf of the tangent bundle which is closed under the Lie bracket,
X
butnotnecessarilyasheafofO -algebras. Whendoinfinitesimaldeformationsoff
X
which are induced by F lift to positive-dimensional deformations of f, where f is
deformedalongthesheafF? InthecasewhereX iscomplex-symplecticandF isthe
sheafoflocallyHamiltonianvectorfields, thispartiallyreproducesknownresultson
unobstructednessofdeformationsofLagrangiansubmanifolds.
MichałKapustkastudiesthevarietyofisotropicfive-spacesofadegeneratefour-
forminaseven-dimensionalvectorspace. Itisadegenerationoftheadjointvarietyof
thesimpleLiegroupG . ItisalsotheimageofP5underthemapinducedbythesystem
2
of quadrics containing a twisted cubic. Using degenerations of this twisted cubic to
threelines,theauthorconstructsgeometrictransitionsbetweenCalabi–Yauthreefolds.
ThearticlebyMateuszMichałekcontainsthenotesfromthelecturesofStefanKe-
bekusonhyperbolicitypropertiesofmodulistacksandgeneralizationsofShafarevich
hyperbolicity to higher dimensions. In the first part, various sheaves of differential
forms on singular algebraic varieties are discussed. The second part collects some
Preface ix
factsontheMinimalModelProgramandlogarithmicsheaves. Thelastpartisdevoted
mainly to hyperbolicity of moduli. It also discusses: pull-backs of differerentials,
a generalization of the Bogomolov–Sommese vanishing, and a special case of the
Lipman–Zariskiconjecture.
MirceaMusta¸ta˘investigatesaninvariantofsingularities,whichplaysanimportant
roleinbirationalgeometry: thelogcanonicalthreshold. Afteritsdefinition,properties
andexamples,theauthordescribesananalogousinvariant: theF-purethreshold,that
comes up in positive characteristic in commutative algebra. Then a sketch of the
proof of Shokurov’sACC conjecture for ambient smooth varieties is presented, and
a connection of this conjecture with Termination of Flips is described. The last part
discussesanasymptoticversionofthelogcanonicalthresholdinthecontextofgraded
sequencesofideals.
The article by Shigeru Mukai concerns K3 surfaces and Enriques surfaces. The
authorstudiestheconnectionbetweensymplecticsymmetriesofK3surfacesandthe
Mathieu group M . He also investigates its Enriques analogy, that is, a conjectural
24
connection between semi-symplectic symmetries of Enriques surfaces and another
MathieugroupM .
12
ÖzerÖztürkandPiotrPragaczstudyThompolynomialsofsingularitiesofmaps,
byexpandingtheminthebasisofSchurfunctions(labelledbypartitions). Thearticle
contains some necessary conditions on a partition to appear in the set of indices of
theSchurfunctionexpansionofaThompolynomial. Moreover,theauthorsdescribe
severalrecursionsforthecoefficientsoftheSchurfunctionsintheseexpansions. The
articlealsoshowsoldandnewcomputationsoftheThompolynomialsofsomesingu-
larities.
Marek Szyjewski investigates the kernel of the norm map on power classes for
cyclic field extensions. In particular, he gives several results on the exactness of the
“generalizedGross–Fischerexactsequence”.
HalszkaTutaj-Gasin´skadiscussesacertainconnectionbetweenthe(local)positivity
oflinebundlesonsmoothprojectivevarietiesandthesymplecticpackingofballsinto
symplectic manifolds. The article collects some results on Seshadri constants and
packingconstants–twokindsofnumericalinvariantsappearinginthisconnection. A
specialemphasisisputonthetoricsituation.
WededicatethewholebooktothememoryofOscarZariski–thefatherofmodern
algebraicgeometry. TheopeningarticlebyPiotrBlassdiscusseshislifeandachieve-
ments. In particular, the influence of Zariski’s work and the work of his students on
contemporaryalgebraicgeometryisemphasized.
ThisvolumecompletestheImpangatrilogywhosefirsttwovolumesare“Topicsin
cohomologicalstudiesofalgebraicvarieties”and“Algebraiccycles,sheaves,shtukas,
andmoduli”,publishedbyBirkhäuserin2005and2008,respectively.
Acknowledgments. The Editor wishes to thank the authors for their contributions.
HeisgratefultoHalszkaTutaj-Gasin´skaforherhelpwiththeImpangaconferencein
Be¸dlewo. Thanks also go to Manfred Karbe and Irene Zimmermann from European
x Preface
MathematicalSocietyPublishingHouseforapleasantcooperationduringtheprepa-
rationofthisbook. TheEditorisgratefulforpartialsupportbyMNiSWgrantNN201
608040duringthepreparationofthisvolume.
Warszawa,June2012 TheEditor