Combinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009

190 Pages·2011·1.81 MB·English

by  Mats Boij

ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
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Gunnar Fløystad (cid:2) Trygve Johnsen (cid:2)
Andreas Leopold Knutsen
Editors
Combinatorial Aspects
of Commutative Algebra
and Algebraic Geometry
The Abel Symposium 2009
Editors
GunnarFløystad TrygveJohnsen
AndreasLeopoldKnutsen UniversityofTromsø
UniversityofBergen Dept.Mathematics&Statistics
Dept.Mathematics 9037Tromsø
JohannesBrunsgate12 Norway
5008Bergen [email protected]
Norway
[email protected]
[email protected]
ISBN978-3-642-19491-7 e-ISBN978-3-642-19492-4
DOI10.1007/978-3-642-19492-4
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011928558
MathematicsSubjectClassification(2010):13-06,14-06
©Springer-VerlagBerlinHeidelberg2011
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Preface to the Series
The Niels Henrik Abel Memorial Fund was established by the Norwegian gov-
ernment on January 1. 2002. The main objective is to honor the great Norwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstanding
scientificworkinthefieldofmathematics.Theprizeshallcontributetowardsrais-
ingthestatusofmathematicsinsocietyandstimulatetheinterestforscienceamong
school children and students. In keeping with this objective the board of the Abel
fundhasdecidedtofinanceoneortwoAbelSymposiaeachyear.Thetopicmaybe
selectedbroadlyintheareaofpureandappliedmathematics.TheSymposiashould
beatthehighestinternationallevel,andservetobuildbridgesbetweenthenational
and international research communities. The Norwegian Mathematical Society is
responsiblefortheevents.Ithasalsobeendecidedthatthecontributionsfromthese
Symposia should be presented in a series of proceedings, and Springer Verlag has
enthusiastically agreed to publish the series. The board of the Niels Henrik Abel
Memorial Fund is confident that the series will be a valuable contribution to the
mathematicalliterature.
RagnarWinther
ChairmanoftheboardoftheNielsHenrikAbelMemorialFund
v

Preface
The title of the Abel Symposium 2009 was “Combinatorial aspects of Commuta-
tiveAlgebraandAlgebraicGeometry”.Thelastcoupleofdecadeshasseenashift
in commutative algebra and algebraic geometry towards more concrete problems,
oftenrelatedtocombinatorics.Theabovewasthereforeanaturalchoicefortheti-
tle of the symposium. In fact many of the topics of the symposium have risen to
prominence only over the last decade. Among those are cluster algebras, tropical
geometry and the recent breakthrough in the studies of syzygies of graded mod-
ules over polynomial rings, called Boij–So¨derberg theory. But also more classical
topics were the subjects of talks, like Hilbert schemes, with the more recent ap-
proach where the parametrized schemes are multigraded, real algebraic geometry,
toric and homogeneous varieties, and cohomology rings and Gromow–Witten in-
variants.Amorerecenttopicisbinomialideals,andtheirprimarydecomposition.
Thepresentvolumeisfortunatetocontainacomprehensiveintroductorysurveyon
thistopicbyEzraMiller,exploringitsapplicationstosuchvariedtopicsashyper-
geometricseries,combinatorialgametheory,andchemicaldynamics.
Thelocalorganizersofthesymposiumwere:
GunnarFløystad,UniversityofBergen,
TrygveJohnsen,UniversityofTromsø,
AndreasL.Knutsen,UniversityofBergen.
Thescientificadvisorycommitteeconsistedof:
AaronBertram,UniversityofUtah,SalkLakeCity,
DavidEisenbud,UniversityofCalifornia,Berkeley,
Frank-OlafSchreyer,UniversityofSaarbru¨cken,
MikeStillman,CornellUniversity,
RaviVakil,StanfordUniversity.
Thesymposiumisintendedtobeasmallerevent,limitedupwardstofortyspeak-
ersandparticipants.Therewerenineteeninvitedspeakers,leadingexpertsintheir
fields.Thespeakersandtitlesofthetalkswereasfollows.
vii
viii Preface
(a) AaronBertram,DeconstructingCoherentSheaves
(b) MatsBoij,HilbertFunctionsandBettiNumbersuptoMultiplesandParameter
Spaces
(c) AndersBuch,QuantumK-TheoryofGrassmannians
(d) AldoConca,SyzygiesofVeroneseandKoszulAlgebras
(e) David Eisenbud, Boij–So¨derberg Theory: Introduction and possible Future
Perspectives
(f) SergeyFomin,EnumerationofPlaneCurvesandLabeledFloorDiagrams
(g) WilliamFulton,CharacterFormulas
(h) Ju¨rgenHerzog,PowersofComponentwiseLinearIdeals
(i) JoelKamnitzer,EquivalencesofDerivedCategoriesfromGeometricsl(2)ac-
tions
(j) DanLaksov,AFormalismforEquivariantCohomology
(k) DianeMaclagen,ConnectedMultigradedHilbertSchemes
(l) EzraMiller,ApplicationsofBinomialCommutativeAlgebra
(m) SamPayne,BoundaryComplexesandWeightFiltrations
(n) Frank-OlafSchreyer,Boij–So¨derbergTheoryforCoherentSheavesonPn
(o) JessicaSidman,SyzygiesoftheSecantVarietiesofCurves
(p) Mike Stillman, Green’s Conjecture and high rank Syzygies on low rank
Quadrics
(q) RekhaThomas,TheConvexHullofaRealAlgebraicVariety
(r) JerzyWeyman,QuiverswithPotentialsandtheirMutations
(s) AndreiZelevinsky,ClusterAlgebrasviaQuiverswithPotentials
The Abel Symposium 2009 was held Monday June 1. to Thursday June 4. in
thetownofVossinWesternNorway.MostofWednesdaywasdevotedtoatripby
bus, boat, and train to the spectacular Nærøyfjord, which is on UNESCO’s world
heritagelist,andtheAurlandsfjord.Inchillybutclearweatherwewentashorefrom
theboattoasmallfarmbytheNærøyfjordandhadanamplelunch.Attheendofthe
AurlandsfjordwetooktheFla˚mrailway,a20kilometertripascending866meterto
Myrdal,andfromtherebacktoVoss.
We would like to express our gratitude to the Abel foundation and the Norwe-
gianMathematicalSocietyforgivingustheopportunitytoorganizetheAbelSym-
posium2009.WealsothanktheadministrationattheDepartmentofMathematics,
University of Bergen for their kind help and assistance with practical matters, we
thankBjørnTheisenfordoingasolidjobwithtypesettingthearticles,andwethank
Springer Verlag and especially Ruth Allewelt for an excellent job in following up
theseproceedings.
BergenandTromsø, GunnarFløystad
22.December2010 TrygveJohnsen
AndreasL.Knutsen
Contents
The Cone of Betti Diagrams of Bigraded Artinian Modules of
CodimensionTwo ............................................... 1
MatsBoijandGunnarFløystad
1 Introduction.............................................. 1
2 Preliminaries ............................................. 3
2.1 BettiDiagramsandtheMultigradedHerzog–Ku¨hl
Equations ........................................ 3
2.2 TheEquivariantResolution ......................... 5
2.3 TheLinearSpaceofBettiDiagramsofMultigraded
ArtinianModules.................................. 5
2.4 TheLinearSpaceintheCaseofTwoVariables......... 6
3 ThePositiveConeofBigradedBettiDiagrams................. 7
3.1 Partitions ........................................ 7
3.2 Decomposing..................................... 9
4 ExistenceofResolutions ................................... 11
5 ResolutionsofTrigradedArtinianModulesofCodimensionThree 14
References..................................................... 16
KoszulCycles................................................... 17
WinfriedBruns,AldoConcaandTimRo¨mer
1 Introduction.............................................. 17
2 NotationandGeneralities .................................. 18
3 BoundsforKoszulCycles .................................. 21
4 Green–LazarsfeldIndexforSegre–VeroneseRings ............. 26
5 GeneratingKoszulCycles .................................. 29
References..................................................... 32
Boij–So¨derbergTheory .......................................... 35
DavidEisenbudandFrank-OlafSchreyer
1 Introduction.............................................. 35
2 BettiTables .............................................. 36
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