Ideals of Powers and Powers of Ideals: Intersecting Algebra, Geometry, and Combinatorics PDF
Preview Ideals of Powers and Powers of Ideals: Intersecting Algebra, Geometry, and Combinatorics
Lecture Notes of the Unione Matematica Italiana Enrico Carlini Huy Tài Hà Brian Harbourne Adam Van Tuyl Ideals of Powers and Powers of Ideals Intersecting Algebra, Geometry, and Combinatorics Lecture Notes of 27 the Unione Matematica Italiana Moreinformationaboutthisseriesathttp://www.springer.com/series/7172 Editorial Board PiermarcoCannarsa FrancoFlandoli (EditorinChief) DipartimentodiMatematica DipartimentodiMatematica Applicata UniversitàdiRoma“TorVergata” UniversitàdiPisa ViadellaRicercaScientifica1 ViaBuonarroti1c 00133Roma,Italy 56127Pisa,Italy e-mail:[email protected] e-mail:[email protected] LuciaCaporaso AngusMaclntyre (Co-EditorinChief) QueenMaryUniversityofLondon DipartimentodiMatematica SchoolofMathematicalSciences UniversitàRomaTre MileEndRoad LargoSanLeonardoMurialdo LondonE14NS,UnitedKingdom I-00146Roma,Italy e-mail:[email protected] e-mail:[email protected] GiuseppeMingione AdolfoBallester-Bollinches DipartimentodiMatematicaeInformatica Departmentd’Àlgebra UniversitàdegliStudidiParma FacultatdeMatemàtiques ParcoAreadelleScienze,53/a(Campus) UniversitatdeValència 43124Parma,Italy Dr.Moliner,50 e-mail:[email protected] 46100Burjassot(València),Spain MarioPulvirenti e-mail:[email protected] DipartimentodiMatematica UniversitàdiRoma“LaSapienza” AnnalisaBuffa PiazzaleA.Moro2 IMATI–C.N.R.Pavia 00185Roma,Italy ViaFerrata1 e-mail:[email protected] 27100Pavia,Italy e-mail:[email protected] FulvioRicci ScuolaNormaleSuperiorediPisa FabrizioCatanese PiazzadeiCavalieri7 MathematischesInstitut 56126Pisa,Italy Universitätstraÿe30 e-mail:[email protected] 95447Bayreuth,Germany e-mail:[email protected] SusannaTerracini UniversitàdegliStudidiTorino CiroCiliberto DipartimentodiMatematica“GiuseppePeano” DipartimentodiMatematica ViaCarloAlberto10 UniversitàdiRoma“TorVergata” 10123Torino,Italy ViadellaRicercaScientifica1 e-mail:[email protected] 00133Roma,Italy e-mail:[email protected] ValentinoTosatti DepartmentofMathematics CorradoDeConcini NorthwesternUniversity DipartimentodiMatematica 2033SheridanRoad UniversitàdiRoma“LaSapienza” Evanston,IL60208,USA PiazzaleAldoMoro5 e-mail:[email protected] 00185Roma,Italy e-mail:[email protected] CorinnaUlcigrai SchoolofMathematics CamilloDeLellis UniversityofBristol SchoolofMathematics 4thFloor,HowardHouse InstituteforAdvancedStudy QueensAvenueBS8 1EinsteinDrive 1SNBristol,UK SimonyiHall e-mail:[email protected] Princeton,NJ08540,USA e-mail:[email protected] TheEditorialPolicycanbefound atthebackofthevolume. ` ` Enrico Carlini • Huy Tai Ha (cid:129) Brian Harbourne (cid:129) Adam Van Tuyl Ideals of Powers and Powers of Ideals Intersecting Algebra, Geometry, and Combinatorics 123 EnricoCarlini HuyTa`iHa` DepartmentofMathematicalSciences DepartmentofMathematics PolitecnicodiTorino TulaneUniversity Torino,Italy NewOrleans,LA,USA BrianHarbourne AdamVanTuyl DepartmentofMathematics DepartmentofMathematicsandStatistics UniversityofNebraska McMasterUniversity Lincoln,NE,USA Hamilton,ON,Canada ISSN1862-9113 ISSN1862-9121 (electronic) LectureNotesoftheUnioneMatematicaItaliana ISBN978-3-030-45246-9 ISBN978-3-030-45247-6 (eBook) https://doi.org/10.1007/978-3-030-45247-6 MathematicsSubjectClassification:13-02,14-02,11P05,14Q05,13D40,13F55,13F20,14C20 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicencetoSpringerNatureSwitzerland AG2020 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Wededicatethisbooktoallofthe participantswho havemadePRAGMATIC suchasuccess for20 years andtothe DepartmentofMathematicsattheUniversity ofCatania,forprovidingsucha supportive researchenvironment. Foreword The PRAGMATIC research school began in 1997 with its first edition. Since the last 20 years, many of the most important researchers in the field of Algebraic GeometryandCommutativeAlgebrahavesucceededasmainteachersofthisproject andhavemadethisschoolarealsuccessamongyoungresearchers(seethewebsite ofPRAGMATIC:www.dmi.unict.it/pragmatic/docs/Pragmatic-main.html). In addition, the almost 400 young researchers who participated in the various editionsofPRAGMATIChaveinturncontributedtothesuccessofthisactivitywith over one hundred publications in the most important scientific journals (a partial list of such works can be found at www.dmi.unict.it/~pragmatic/docs/Pragmatic- papers.html).Inthanking,onbehalfoftheorganizers,allthosewhocontributedto theenormoussuccessofPRAGMATICattheendofthesefewlines,thelistofmain speakers and collaborators who have been present over the years is included. To makethisschoolofresearchevenmorepopular,the2017editionhasbeenenriched by the present nice volume that collects the lessons held during the period of the school and all the material necessary to address the open problems proposed in this edition. The authors, all of whom were involved in Pragmatic 2017, had the wonderfulideaofputtingtogetherboththepreparatorymaterial,thelessonsmade duringtheschoolperiod,theopenproblemsassignedandthestateoftheartatthe endofthethreeweeksofactivity. Already the intriguing title announcesin an original and curious way the field of research that is treated. Precisely, in this book powers of ideals and ideals of powersareapproachedfromthreedifferentpointsofview:algebra,combinatorics, and geometry, with special regard to the interactions among these perspectives. I believethatthese noteswillbe usefulnotonlyforthegroupof participantsin the 2017editionofPRAGMATICbutalsoforallthosewhohaveinterestsinthestudy of the powers of ideals and their applications in different fields of mathematics. Moreover,thewaythetextiscomposedalsoprovidesasimpleandfruitfulwayto approachopenissuesinthisarea.TheorganizersofPRAGMATICareverygrateful to Enrico Carlini, Huy Tài Hà, Brian Harbourne,and Adam Van Tuyl for making vii viii Foreword this additional effort to be useful to the young mathematical community and for doingsointhecontextofPRAGMATIC. Here is the list (in a chronological order) of the main teachers involved in all the editions of PRAGMATIC: David Eisenbud, Lawrence Ein, Anthony V. Geramita, Juan Migliore, Klaus Hulek, Kristian Ranestad, Ciro Ciliberto, Rick Miranda, Fyodor Zak, Massimiliano Mella, Igor Dolgachev, Alessandro Verra, Olivier Debarre, Lucia Caporaso, Lucian Badescu, Francesco Russo, Giuseppe Pareschi,MihneaPopa,JurgenHerzog,VolkmarWelker,RosaMiróRoig,Giorgio Ottaviani,PaltinIonescu,JaroslavA.Wisnieswski,RalfFröberg,MatsBoij,Alessio Corti, Paolo Cascini, Yujiro Kawamata, Aldo Conca, Srikanth Iyngar, Anurag Singh,AlessandroChiodo,FilippoViviani,GianPietroPirola,JoanCarlesNaranjo, BrianHarbourne,andAdamVanTuyl. Andhereisalistofyoungcollaboratorsinvolvedduringtheseyears:S.Popescu, V.Masek,A.Bigatti,C.Peterson,F.Flamini,A.Bruno,G.Pacienza,C.Casagrande, A.Rapagnetta,M.Vladiou,X.Zheng,E.Nevo,L.Costa,D.Faenzi,L.SolaCondé, J.C.Sierra,V.Crispin,A.Engström,A.Kasprzyk,Y.Gongyo,G.Codogni,J.Guéré, L.Stoppino,V.Gonzalez-Alonso,E.Carlini,andHuyTàiHà. Catania,Italy AlfioRagusa September2018 Preface This book contains reorganized and extended versions of our lectures at PRAG- MATIC 2017, held from June 19th to July 7th, 2017. PRAGMATIC (Promotion of Research in Algebraic Geometry for MAThematicians in Isolated Centres) is an annual summer school, started in 1997 and organized by the Università di Catania,Catania,Italy.Thegoaloftheschoolistostimulateresearchinalgebraic geometryamongPh.D.studentsandearlycareerresearchers,especiallythoseliving inisolatedcentersorperipheraluniversitiesalloverEurope. We celebrate PRAGMATIC’s twentieth anniversary with the theme “powers of ideals and ideals of powers.” This theme became the title of our book. The topicsin this bookare motivatedby algebraicproblemsinvolvingthe relationship between various notions of powers of ideals and by related geometric problems. Thisincludes,asjustoneexampleamongaconstellationofvariations,theWaring problemofwritingahomogeneouspolynomialasaminimalsumofpowersoflinear homogeneous polynomials, which translates algebraically into studying ideals of powersandgeometricallyintostudyingdimensionsofsecantvarieties. In these notes, powers of ideals and ideals of powers are approached from threepointsof view—algebra,combinatorics,andgeometry—andtheinteractions between these perspectives will be developed. Readers are invited to explore the evolutionof the set of associated primes of higher and higher powersof an ideal. For ideals associated with a combinatorialobject like a graph or hypergraph,one wishes to explain this evolution in terms of the original combinatorial objects. Similar questions concern understanding the Castelnuovo–Mumford regularity of powers of combinatoriallydefined ideals in terms of the associated combinatorial data. From a more geometric point of view, one can consider how the relations betweensymbolicandregularpowerscanbeinterpretedingeometricalterms.Other topics to be presented include aspects of Waring type problems, symbolic powers ofanidealandtheirinvariants(e.g.,theWaldschmidtconstant,theresurgence),and thepersistenceofassociatedprimes. ix x Preface When preparing our lectures for PRAGMATIC, our emphasis was on quickly introducingtheparticipantstoopenproblemsandquestionsintheseresearchareas. At the same time, we wantedto providerestricted versionsof these problemsand questionsfocusedonspecificcaseswhichparticipants,withaminimalbackground intheseareas,couldtackle.Ourintentionwasforthesespecificcasestobesimple enoughthatparticipantswouldmakesignificantprogresswithinthe3-weekschool timeframe,andyetimportantenoughthattheirworkcouldleadtopublicationsand furtherinvestigationoftheseproblemsandquestionsinmoregenerality.Withthis inmind,ourfocuswasonthecontextoftheproblemsandonhowproblems,results, andmethodshaveevolved.Consequently,ourlecturenotesoftenomitthedetailed proofsofstatedtheoremsorjustsketchoutimportantideas. The book is divided into six parts. In the first part, we discuss the associated primesof idealsand, in particular,the persistence propertyand the stability index of these sets. In the second part, we investigate the asymptotic linearity of the Castelnuovo–Mumfordregularityof powersof a homogeneousideal. Most of our attentionwillberestrictedtosymbolicandordinarypowersofedgeidealsofgraphs. The third part of the book is devoted to the containments between symbolic and ordinarypowersofideals,focusingonsquarefreemonomialidealsandthedefining idealsofschemesoffatpoints.InPartIV,weexaminetheveryrecentlyintroduced notion of unexpected curves and the role of the SHGH conjecture in inspiring it. In Part V, we discuss the Waring problem for homogeneous polynomials. Specifically,wedescribeSylvester’salgorithmforbinaryformsandtheconnection toStrassen’sconjecture.PartVIofthebookisasummaryofmaterialspresentedat thePRAGMATICschool.Inparticular,wehaveincludedachapteron“TheArtof Research,”whichaimsathelpingyoungresearcherswithadviceonhowto starta researchproject,onhowtocollaborate,onhowtowriteuptheirresults,andonhow topresenttheirfindings. We assume that the interested reader is familiar with basic concepts from commutative algebra. Unexplained notations and terminology can be found in standardtexts[14,25,47,63,131,137,155,166]. Torino,Italy EnricoCarlini NewOrleans,LA,USA HuyTàiHà Lincoln,NE,USA BrianHarbourne Hamilton,ON,Canada AdamVanTuyl July2017–December2019
The list of books you might like
