Bridging Algebra, Geometry, and Topology

295 Pages·2014·2.29 MB·English

by  Denis Ibadula

Springer Proceedings in Mathematics & Statistics
Volume 96
Moreinformationaboutthisseriesathttp://www.springer.com/series/10533
Springer Proceedings in Mathematics & Statistics
Thisbookseriesfeaturesvolumescomposedofselectcontributionsfromworkshops
and conferences in all areas of current research in mathematics and statistics,
includingORandoptimization.Inadditiontoanoverallevaluationoftheinterest,
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individual contributions are all refereed to the high quality standards of leading
journals in the field. Thus, this series provides the research community with
well-edited, authoritative reports on developments in the most exciting areas of
mathematicalandstatisticalresearchtoday.
Denis Ibadula • Willem Veys
Editors
Bridging Algebra, Geometry,
and Topology
123
Editors
DenisIbadula WillemVeys
FacultyofMathematics DepartmentofMathematics
andInformatics UniversityofLeuven
OvidiusUniversity Leuven(Heverlee),Belgium
Constanta,Romania
ISSN2194-1009 ISSN2194-1017(electronic)
ISBN978-3-319-09185-3 ISBN978-3-319-09186-0(eBook)
DOI10.1007/978-3-319-09186-0
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014950409
MathematicsSubjectClassification(2010):13-XX,14-XX,51-XX,54-XX
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Preface
Thepresentvolumecontainsrefereedpaperswhichwerepresentedatthe Interna-
tional Conference “Experimental and Theoretical Methods in Algebra, Geometry
andTopology,”heldinEforieNord(nearConstanta),Romania,during20–25June
2013. The conference was devoted to the 60th anniversary of two distinguished
Romanian mathematicians: Alexandru Dimca from the Université de Nice-Sophia
Antipolis,FranceandS¸tefanPapadimafromtheInstituteofMathematics“Simion
Stoilow”oftheRomanianAcademy.
Theconferencebroughttogethermorethan80expertsintopology,singularities,
hyperplane arrangements, combinatorics, algebraic geometry, and commutative
algebra from almost all continents of the world. It aimed to strengthen regional
andprofessionalnetworkingbetweenhighlevelspecialistsfromdifferentfieldsof
mathematics.
This meeting had a special significance for the community of mathematicians
from Romania;it was dedicated to a Romanian mathematician from diaspora and
a “local” one, and many other top Romanian mathematicians, living and working
abroad,honoredtheinvitationtoparticipateintheconference.
Algebra, geometry, and topology cover a variety of different, but intimately
related, research fields in modern mathematics. This book focuses on specific
aspectsofthisinteraction.
Theselectedpaperscontainoriginalresearchworkandasurveypaper.Theyare
intendedforalargeaudience,includingresearchersandgraduatestudentsinterested
in algebraic geometry, combinatorics, topology, hyperplane arrangements, and
commutativealgebra.Thepapersarewrittenbywell-knownexpertsfromdifferent
fields of mathematics, affiliated to universities from all over the word; they cover
a broad range of topics and explore the research frontiers of a wide variety of
contemporaryproblemsofmodernmathematicsthatarerarelybroughttogetherin
thismanner.
Now,wedescribebrieflythecontentofeachpaperfromthisvolume.
v
vi Preface
Solutions of a zero-dimensional system of multivariate polynomials over the
rationals could be found using the so-called triangular decomposition. Important
work on this subject was done by Möller, who has also given an algorithm to
compute the triangular decomposition. The purpose of the paper “Solving via
Modular Methods” by Deeba Afzal, Faira Kanwal Janjua, Gerhard Pfister, and
Stefan Steidel is to present a parallel modular algorithm in the line of Moreno
Maza–Xie and Li–Moreno Maza with focus on a probabilistic algorithm. Due to
the presentarchitectureof the processors,the parallelalgorithmsare an important
researchdirectioninthisfield.
The paperofMarianAproduon“Lazarsfeld–MukaiBundlesandApplications:
II”isaniceshortexpositionofthetheoryofLazarsfeld–Mukaibundlesonsurfaces,
providingsome improvementsof knownresultsin the case of rationalsurfacesas
well. The author explains how these bundles can be used in order to bound the
dimensionofBrill–Noetherloci,whichisinturnusefulinaddressingproblemson
syzygies,likethefamousGreenconjecture.Thepaperconcludeswithasolutionof
thisconjectureforsomespecialcurvesonspecialrationalsurfaces.
Multinets are of interest for their relationship with the fundamental group
of a complex hyperplane arrangement complement and the cohomology of one-
dimensionallocal systems. If all multiplicities are equal to 1, then a multinet is a
netthatisarealizationbylinesandpointsofseveralorthogonalLatinsquares.Very
few examples of multinets with nontrivial multiplicities are known. In the paper
“Multinetsin P2,” JeremiahBartz andSergeyYuzvinskypresentnew examplesof
multinets. These are obtainedby using an analogue of nets in P3 and intersecting
them by planes. Prior to the result in this paper, there were almost no systematic
constructionsof infinite families of multinets. This paper gives a new and natural
construction,certaintobeoffurtherinterest.
The coGalois theory studies the correspondence between subfields of a rad-
ical field extension L=K and subgroups of the coGalois group coG.L=K/ WD
Tors.L(cid:2)=K(cid:2)/. In the paper“A More GeneralFramework for CoGaloisTheory,”
S¸erbanBasarabcontinueshisworkonabstractco-Galoistheorywhichconcernsa
continuousaction(cid:2)(cid:2)A!Aofaprofinitegrouponadiscretequasi-cyclicgroup
A. The author investigates Kneser triples and Cogalois triples providing general
Kneser andCogaloiscriteria.He states problemsonthe classification of the finite
structure,arisingnaturallyfromthesecriteria.
LetP beaproductofweightedprojectivespaces.Inhispaper“Connectivityand
aProblemofFormalGeometry,”LucianBa˘descuprovesanalgebraizationresultfor
formal-rationalfunctionsoncertainclosedsubvarietiesX ofP (cid:2)P.Firstofall,the
maintheoremofthepaperisanimprovementofaconnectivityresultofL.BaMdescu
andF.Repetto.Secondly,themainresultofthepaperhastwointerestingcorollaries.
The first one extends to arbitrary characteristic and weighted projective spaces a
resultobtainedincharacteristic0andordinaryprojectivespacebyFaltings,andthe
lastoneextendsalsototheweightedcasearesultofFaltings.
The paper “Hodge Invariants of Higher-Dimensional Analogues of Kodaira
Surfaces”ofVasile Brînzaˇnescudealswitha classof compactcomplexmanifolds
whichappearastorus-bundlesoveranellipticcurve.Whentheyarenon-Kählerian,
Preface vii
they are called Kodaira manifolds since they are generalizations of the Kodaira
surfaces to higher dimension. The main contributions of the paper are the com-
putation of their Hodge numbers and establishing the existence of a holomorphic
symplectic structure in the even dimensional case. The author thus brings further
examples of holomorphic symplectic manifolds, which were recently intensively
studied,althoughinadifferentsetup.
In “An Invitation to Quasihomogeneous Rigid Geometric Structures,” Sorin
Dumitrescusurveyshisresultsonquasihomogeneousrigidgeometricstructuresof
manifolds that encompass more than a decade of work. His results take place in
the real or complex setting, sometimes concern general geometric structures, and
sometimes concern affine connections or Lorenzian metrics. The problems that
have been studied by Dumitrescu concern geometric structures whose isometry
pseudogroup has big orbits, and the extent to which this already big orbit is all
ofthemanifold,touchinguponmanypartsofgeometry.
Thefundamentalgroupofthecomplementofasingularplanecurve,introduced
andstudiedbyZariskiasanapproachtothedeformationclassificationofsurfaces
ofgeneraltype,hasbeena subjectofcloseinterestsincethen.Usually,thisgroup
isverydifficulttocompute,anditisequallydifficulttoproduceexamplesofcurves
with interesting groups. In “On the FundamentalGroups of Non-generic R-Join-
Type Curves,” Christophe Eyral and Mutsuo Oka use the concept of bifurcation
graph (a version of Grothendieck’s dessins d’enfants, developed by the authors
earlier)andshowthatifthecurveissufficientlygeneric,thenitsfundamentalgroup
isstilltheminimalgroupdeterminedbythemultiplicitiesoftheroots.
Motivated by questions in algebraic statistics, there is interest to study the
binomialedgeidealofafinitesimplegraphG.Inthepaper“KoszulBinomialEdge
Ideals,”VivianaEne,JürgenHerzog,andTakayukiHibishowthatif thebinomial
edge ideal of G defines a Koszul algebra, then G must be chordaland claw free.
Aconverseofthisstatementisprovedforaclassofchordalandclawfreegraphs.
In the paper “Some Remarks on the Realizability Spacesof (3, 4)-Nets,” Denis
Ibadula and Anca Daniela Ma˘cinic prove that in the class of .3;4/-nets with
doubleandtriplepoints,latticeisomorphismactuallytranslatesintolatticeisotopy.
Moreover, they disprove the existence of Zariski pairs involving an example of
Yoshinagaofa.3;6/-netwith48triplepoints.
ThepaperofS¸tefanPapadimaandAlexanderI.Suciu,“Non-AbelianResonance:
ProductandCoproductFormulas,”isaboutalgebraicaspectsofresonancevarieties.
Originally, the resonance varieties appeared as a natural way to put together
information about wedge products of the cohomology of a topological space X
with 1-cohomology classes. This is the so-called formal rank 1 case, or, almost
equivalently, the “formal abelian” case. In this paper, more general cases are
consideredbydroppingoneorbothoftheassumptions.Inotherwords,thehigher
ranklocalsystemsareconsideredandpossiblynon-formalsituations.Theauthors
addressalgebraically,viageneralizedresonancevarieties,thesituationsarisingfrom
the product and the wedge of two topological spaces. They thus address basic
viii Preface
foundational questions and obtain concrete results, which will definitely be used
in futureresearchin thisdirection.Inparticulartheydescribepreciselytherank2
formalcase.
The paper “Gauss–Lucas and Kuo–Lu Theorems” of Lauren¸tiu Pa˘unescu is a
nice, short paper which contains elementary observationson the classical Gauss–
LucasTheoremandtheKuo–Lutheorem.Anelementaryalgebraiccalculationover
the Newton–Puiseux field, only employing its contact order structure, shows that
the Kuo–Lu theorem is in fact a Gauss–Lucas type theorem, via a new notion of
convexityovertheNewton–Puiseuxfield.
The germ of a plane curve has a minimal desingularization, constructed by
blowing up points. The number of blow-ups is an invariant, called blow-up
complexity.MaríaPePereiraandPatrickPopescu-Pampu,in“FibonacciNumbers
andSelf-DualLatticeStructuresforPlaneBranches,”relatetheblow-upcomplexity
withthemultiplicityandtheMilnornumber.Fibonaccinumbersappearnaturallyin
thisstudy.Theproofsarecombinatorial,basedonEnriquesdiagrams.Theauthors
constructapartialorderandanaturaldualityonthesetofEnriquesdiagramswith
fixed blow-up complexity. The duality is interesting and differs from projective
dualityingeneral.
Stanley’s conjecture on multigraded modules over the polynomial ring S D
KŒx ;(cid:3)(cid:3)(cid:3) ;x (cid:3),whereK isafield,hasattractedmanyresearchersinthelastdecade
1 n
and is still widely open. Some progress has been done by Dorin Popescu in a
seriesofworkswhichtreatStanley’sconjectureorweakerformsofitforsquarefree
multigradedmodulesoftheformI=J whereI (cid:4)J aresquarefreemonomialideals
inS.Inthepaper“FourGenerated,Squarefree,MonomialIdeals,”AdrianPopescu
andDorinPopescushowthatunderrestrictedconditionsonthegeneratorsofI and
J,iftheStanleydepthofI=J isd C1,thendepthI=J (cid:5)d C1.HenceStanley’s
conjectureholdsinthiscase.
The aim of the paper of J.H.M. Steenbrink, “Motivic Milnor Fibre for Nonde-
generateFunctionGermsonToricSingularities,”istogiveacombinatorialformula
forthemotivicMilnorfiberofanondegeneratefunctiongermonatoroidalvariety.
The paperprovidesa nice introductionto the theoryof motivicnearbycyclesand
the use of toroidal methods. The final expression of the given formula contains
certainclassesofvarietiesdefinedbyanequationoftheformg D 1withg quasi-
homogeneous.
Intheirpaper“TheConnectedComponentsoftheSpaceofAlexandrovSurfaces,”
Joël Rouyer and Costin Vîlcu study the connected components of the space of
(pairwisenon-isometric)Alexandrov(two-dimensional)surfaces.DenotebyA.k/
the set of all compact Alexandrov surfaces without boundary with curvature
boundedbelowbyk,endowedwiththetopologyinducedbytheGromov–Hausdorff
metric.TheydeterminetheconnectedcomponentsofA.k/andofitsclosure.
In his paper “Complements of Hypersurfaces, Variation Maps and Minimal
Models ofArrangements,” Mihai Tiba˘r provesthe minimality of the CW-complex
structure for complements of hyperplane arrangements in Cn. The proof of this
importantresult, originallydueto Randelland Dimca–Papadima,is obtainedhere
usingLefschetzpencilsandvariationmapswithinapencilofhyperplanes.Thereis
Preface ix
arenewedinterestinthetechniquesusedinthispaper,whicharedevelopedbythe
authorinalongseriesofpapers,becauseofthecentralroletheyplayedintherecent
proofofaconjectureofDimca–PapadimabyHuh.
Finally,thepaper“CriticalPointsofMasterFunctionsandthemKdVHierarchy
of Type A.2/” deals with m-parameter families of critical points of the master
2
functionassociatedwith the trivialrepresentationofthe twisted affineLie algebra
A.2/. Alexander Varchenko, Tyler Woodruff, and Daniel Wright show that the
2
embeddingofthefamilyintothespaceMoftheMiuraopersoftypeA.2/definesa
2
varietywhichisinvariantwithrespecttoallmKdVflowsonM,andthatthisvariety
ispoint-wisefixedbyallflowsofbigenoughindex.
TheconferencewasorganizedbyOvidiusUniversityofConstanta,incoopera-
tionwiththeInstituteofMathematics“SimionStoilow”oftheRomanianAcademy
(IMAR)andtheRomanianMathematicalSociety,anditwasasatelliteconference
of the Joint InternationalMeeting of the American Mathematical Society and the
RomanianMathematicalSociety,heldin2013inAlbaIulia,Romania.
The scientific committee of our conference was formed by Dan Burghelea
(Ohio State University, USA), Octav Cornea (Centre de Recherches Mathéma-
tiques, Canada), Ezra Miller (Duke University, USA), Andras Némethi (Rényi
Mathematical Institute, Hungary), Claude Sabbah (Ecole Polytechnique, France),
Bernd Sturmfels (University of California, USA), Alexander Suciu (Northeastern
University,USA),andWimVeys(UniversityofLeuven,Belgium).
The organizingcommittee was formed by Marian Aprodu (IMAR), Wladimir-
Georges Boskoff (Ovidius University), Viviana Ene (Ovidius University), Denis
Ibadula(OvidiusUniversity),AncaMaMcinic(IMAR),NicolaeManolache(IMAR),
andAlexanderSuciu(NortheasternUniversity).
The organizers would like to thank the Foundation Compositio Mathematica
for providing a significant financial support for organizing this conference. We
also gratefully acknowledge the support offered by our sponsors CNRS Franco-
Romanian LEA Mathématiques et Modélisation, Institut Universitaire de France,
andCNCSGrantPN-II-PCE-2011-3-028,RomanianMathematicalSociety.
Specialthankstoallcontributorsforthequalityoftheirpapersandtoallreferees
for contributing to the improved scientific level of these proceedings. We would
also like to thank Ms. J. Mary Helena, Project manager at Spi Global, Dr. Eve
Mayer,Assistant EditorMathematicsfrom SpringerScience and Business Media,
andtheSpringerproductionteamfortheirpatientguidanceinthepreparationofthis
volume.
Onbehalfofallthecontributorsandtheorganizersoftheconference,wededicate
this volume to the outstanding mathematicians, mentors, colleagues, and friends
AlexandruDimcaandS¸tefanPapadima,ontheoccasionoftheir60thanniversary!
Constanta,Romania DenisIbadula
Leuven,Belgium WimVeys