Algebraic Curves: Towards Moduli Spaces

237 Pages·2019·4.101 MB·English

by  Kazaryan, Maxim E., Lando, Sergei K., Prasolov, Viktor Vasilʹevich

Moscow Lectures 2
Maxim E. Kazaryan · Sergei K. Lando
Victor V. Prasolov
Algebraic
Cur ves
Towards Moduli Spaces
Moscow Lectures
Volume 2
SeriesEditors
LevD.Beklemishev,Moscow,Russia
VladimirI.Bogachev,Moscow,Russia
BorisFeigin,Moscow,Russia
ValeryGritsenko,Moscow,Russia
YulyS.Ilyashenko,Moscow,Russia
DmitryB.Kaledin,Moscow,Russia
AskoldKhovanskii,Moscow,Russia
IgorM.Krichever,Moscow,Russia
AndreiD.Mironov,Moscow,Russia
VictorA.Vassiliev,Moscow,Russia
ManagingEditor
AlexeyL.Gorodentsev,Moscow,Russia
Moreinformationaboutthisseriesathttp://www.springer.com/series/15875
Maxim E. Kazaryan • Sergei K. Lando (cid:129)
Victor V. Prasolov
Algebraic Curves
Towards Moduli Spaces
123
MaximE.Kazaryan SergeiK.Lando
SteklovMathematicalInstituteofRAS NationalResearchUniversityHigher
NationalResearchUniversityHigher SchoolofEconomics
SchoolofEconomics SkolkovoInstituteofScience
SkolkovoInstituteofScience andTechnology
andTechnology Moscow,Russia
Moscow,Russia
VictorV.Prasolov
IndependentUniversityofMoscow
Moscow,Russia
TranslatedfromtheRussianbyNataliaTsilevich.:OriginallypublishedasАлгебраические
Кривыепо направлениюк пространствам модулейbyMCCME,2018.
ISSN2522-0314 ISSN2522-0322 (electronic)
MoscowLectures
ISBN978-3-030-02942-5 ISBN978-3-030-02943-2 (eBook)
https://doi.org/10.1007/978-3-030-02943-2
LibraryofCongressControlNumber:2018964412
MathematicsSubjectClassification(2010):14H10,14H37,14H55,14H70
©SpringerNatureSwitzerlandAG2018
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Preface to the Book Series Moscow
Lectures
YouholdavolumeinatextbookseriesofSpringerNaturededicatedtotheMoscow
mathematical tradition. Moscow mathematics has very strong and distinctive fea-
tures.Thereareseveralreasonsforthis,allofwhichgobacktogoodandbadaspects
of Soviet organization of science. In the twentieth century, there was a veritable
galaxyofgreatmathematiciansinRussia,whileitsohappenedthattherewereonly
few mathematicalcenters in which these experts clustered. A major one of these,
andperhapsthemostinfluential,wasMoscow.
There are three major reasons for the spectacular success of Soviet mathemat-
ics:
1. Significant support from the governmentand the high prestige of science as a
profession.Bothfactorswererelatedtotheprocessofrapidindustrializationin
theUSSR.
2. Doing research in mathematics or physics was one of very few intellectual
activities thathad nomandatoryideologicalcontent.Many would-becomputer
scientists,historians,philosophers,oreconomists(andevenartistsormusicians)
becamemathematiciansorphysicists.
3. TheIronCurtainpreventedinternationalmobility.
These are specific factors that shaped the structure of Soviet science. Certainly,
factors(2)and(3)aremoreonthenegativesideandcannotreallybecalledfavorable
but they essentially came together in combination with the totalitarian system.
Nowadays, it would be impossible to find a scientist who would want all of the
threefactorstobebackintheirtotality.Ontheotherhand,thesefactorsleftsome
positiveandlonglastingresults.
An unprecedented concentration of many bright scientists in few places led
eventuallytothedevelopmentofaunique“Sovietschool”.Ofcourse,mathematical
schools in a similar sense were formed in other countries too. An example is the
Frenchmathematicalschool,whichhasconsistentlyproducedfirst-rateresultsover
alongperiodoftimeandwhereanextensivedegreeofcollaborationtakesplace.On
the other hand, the British mathematicalcommunitygave rise to many prominent
successes but failed to form a “school” due to a lack of collaborations. Indeed, a
v
vi PrefacetotheBookSeriesMoscowLectures
school as such is not only a large group of closely collaborating individuals but
alsoagroupknittightlytogetherthroughstudent-advisorrelationships.IntheUSA,
whichiscurrentlytheworldleaderintermsofthelevelandvolumeofmathematical
research, the level of mobility is very high, and for this reason there are no US
mathematical schools in the Soviet or French sense of the term. One can talk not
only about the Soviet school of mathematics but also, more specifically, of the
Moscow, Leningrad, Kiev, Novosibirsk, Kharkov, and other schools. In all these
places, there were constellationsof distinguished scientists with large numbersof
students,conductingregularseminars.Thesedistinguishedscientistswereoftennot
merelyadvisorsandleaders,butoftentheyeffectivelybecamespiritualleadersina
verygeneralsense.
AcharacteristicfeatureoftheMoscowmathematicalschoolisthatitstressesthe
necessity for mathematicians to learn mathematics as broadly as they can, rather
thanfocusingonanarrowfieldinordertogetimportantresultsassoonaspossible.
The Moscow mathematical school is particularly strong in the areas of alge-
bra/algebraicgeometry,analysis,geometryandtopology,probability,mathematical
physics and dynamical systems. The scenarios in which these areas were able to
developinMoscowhavepassedintohistory.However,itispossibletomaintainand
develop the Moscow mathematical tradition in new formats, taking into account
modern realities such as globalization and mobility of science. There are three
recently created centers—the Independent University of Moscow, the Faculty of
Mathematics at the National Research University Higher School of Economics
(HSE) and the Center for Advanced Studies at Skolkovo Institute of Science and
Technology(SkolTech)—whosemissionistostrengthentheMoscowmathematical
traditioninnewways.HSEandSkolTechareuniversitiesofferingofficiallylicensed
full-timeeducationalprograms.Mathematicalcurriculaattheseuniversitiesfollow
not only the Russian and Moscow tradition but also new global developments in
mathematics. Mathematical programs at the HSE are influenced by those of the
Independent University of Moscow (IUM). The IUM is not a formal university;
it is rather a place where mathematicsstudentsof differentuniversitiescan attend
specialtopicscoursesaswellascourseselaboratingthecorecurriculum.TheIUM
was the main initiator of the HSE Faculty of Mathematics. Nowadays, there is a
closecollaborationbetweenthetwoinstitutions.
While attempting to further elevate traditionally strong aspects of Moscow
mathematics, we do not reproduce the former conditions. Instead of isolation
and academic inbreeding, we foster global sharing of ideas and international
cooperation. An important part of our mission is to make the Moscow tradition
ofmathematicsatauniversitylevelapartofglobalcultureandknowledge.
The“MoscowLectures”seriesservesthisgoal.Ourauthorsaremathematicians
of different generations. All follow the Moscow mathematical tradition, and all
teach or have taught university courses in Moscow. The authors may have taught
mathematics at HSE, SkolTech, IUM, the Science and Education Center of the
Steklov Institute, as well as traditionalschools like MechMath in MGU or MIPT.
Teaching and writing styles may be very different. However, all lecture notes are
PrefacetotheBookSeriesMoscowLectures vii
supposedtoconveyalivedialogbetweentheinstructorandthestudents.Notonly
personalitiesofthelecturersareimprintedinthesenotes,butalsothoseofstudents.
We hope that expositions published within the “Moscow lectures” series will
provideclearunderstandingofmathematicalsubjects,usefulintuition,andafeeling
oflifeintheMoscowmathematicalschool.
Moscow,Russia IgorM.Krichever
VladlenA.Timorin
MichaelA.Tsfasman
VictorA.Vassiliev
Introduction
The foundations of the theory of Riemann surfaces were laid in the second half
of the nineteenth century. It became the convergence point of the then advanced
developmentsinanalysis,algebra,andyet-to-be-createdtopology.
Duringthewholetwentiethcentury,thetheoryofRiemannsurfaces,mergedwith
thetheoryofcomplexalgebraiccurves,wouldrepeatedlymoveintotheforeground
of mathematics. It explained many difficulties arising when integrating various
functions and made it possible to develop efficient integration methods, clarified
Galois theory and led to a new understanding of arithmetics and number theory,
becameatestinggroundforthetheoryofcomplexvarietiesandfunctionalclasses.
In the last decades of the twentieth century, Riemann surfaces and their moduli
spacesprovedtobemuchneededasoneofthemostefficienttoolsinthestudyof
integrablesystems andrelated modelsof mathematicalphysics.It is these aspects
ofthetheoryofRiemannsurfacesthatarethemainsubjectofthebook.
The principal feature of modern applications is that they use not individual
curves, but families of curves depending analytically on a parameter. For this
reason, the book pays much attention to those properties of individual curves
that play a crucial role when passing to families of curves, such as, for example,
the existence of nontrivial symmetries. Curves with nontrivial symmetries form
naturalsubvarietiesinmodulispacesofcurvesand,ina sense,determineorbifold
characteristicsofthesespaces.Mostmonographsandtextbooksonalgebraiccurves
either work with individual curves and are fairly elementary (like, for example,
[6,16],or[5]),orreckononusingadvancedalgebro-geometrictechniquesandthus
requireasolidbackground(like[11]or[2]).Thepurposeofthisbookistofillthe
existinggapwhileremainingintheframeworkofmoreorlesselementarymethods,
and providesenior students with a tool for dealing with modulispaces. We’d like
tohopethatphysicists,forwhommodulispacesofcurvesareatoolratherthanan
objectofstudy,willalsofindhereanswerstosomenaturalquestionswithoutgoing
intotechnicalities.
Themainpropertiesofmodulispacesofcurvesandmapsinterestingforappli-
cationsaretheirtopologicalcharacteristics.Thesecharacteristicscanbeexpressed
primarily in terms of vector bundles over moduli spaces and their characteristic
ix
x Introduction
classes. When presenting this material, we tried not to overstepthe boundsof the
theory of Chern classes of vector bundles and demonstrate only calculations that
can be performedmanually.There is no doubtthatthe topologyof modulispaces
of curvesof higher genusis extremelycomplicated,and one can hardly expectto
obtain its complete description, at least in the near future. However, information
on these spaces needed for applications (“tautological cohomology ring,” defined
one way or another)can proveto be muchmore available.In thisrespect, moduli
spacesofrationalcurveswithmarkedpointscanserveasagoodtestinggroundfor
developing an intuition for the structure of more complicated spaces, so they are
coveredinaseparatechapter.InthischapterwediscussKapranov’smodel,which
representsstablerationalcurvesasVeronesecurvesinappropriateprojectivespaces,
aswellasKeel’sandKontsevichandManin’stheoremsdescribingthecohomology
ofmodulispacesofrationalcurves.
TheintroductiontoGraphsonSurfacesandTheirApplicationsbyS.K.Lando
andA.K.Zvonkin(Springer-Verlag,Berlin–Heidelberg,2004)says:
Itisclearthateverychapterofourbookcouldgiverisetoanentirebook.Probably,oneday
suchaseriesofbookswillbewritten.
Atleastoneoftheauthorsofthepresentbookregardsitasanelementofthisfuture
series, namely, as the background for Chaps.4 and 5 in Graphs on Surfaces. In
turn,thelatterbookcontainsa fairamountofinformationonmodernapplications
ofalgebraiccurvesandtheirmodulispaces.
The present book is based on a 1-year course taught for a number of years at
theIndependentUniversityofMoscow,andthenattheDepartmentofMathematics
of the Higher School of Economics. The course is intended for third and fourth
yearundergraduatesandgraduatestudentswithnopreviousknowledgeofalgebraic
curves.Aregularlecturecourserequiresalargenumberofexercises,inparticular
intended for the evaluation of student learning. For the reader’s convenience, the
exercisesscatteredthroughoutthebookarealsocollectedintheconcludingchapter.
Thischapteralsocontainssampleexamproblemsandquestions.
Whileworkingonthebook,M.E.KazaryanandS.K.Landoweresupportedby
theRSFgrant16-11-10316Characteristicclassesandrepresentationtheory.