Ideals of Powers and Powers of Ideals: Intersecting Algebra, Geometry, and Combinatorics

162 Pages·2020·2.138 MB·English

by  Enrico Carlini

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
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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
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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