Cohomology of arrangements and moduli spaces [thesis]

227 Pages·2016·1.404 MB·English

by  Olof Bergvall

Cohomologyofarrangementsandmodulispaces
OlofBergvall

Cohomology of arrangements and
moduli spaces
Olof Bergvall
©OlofBergvall,Stockholm,2016
Address:MatematiskaInstitutionen,StockholmsUniversitet,10691Stockholm
E-mailaddress:[email protected]
ISBN978-91-7649-489-9
PrintedbyHolmbergs,Malmö,2016
Distributor:Publit
Abstract
This thesis mainly concerns the cohomology of the moduli spaces M [2]
3
and M [2] of genus 3 curves with level 2 structure without respectively
3,1
withamarkedpointandsomeoftheirnaturalsubspaces. Agenus3curve
whichisnothyperellipticcanberealizedasaplanequarticandthemoduli
spacesQ[2]andQ [2]ofplanequarticswithoutrespectivelywithamarked
1
pointaregivenspecialattention. Thespacesconsideredcomewithanat-
uralactionofthesymplecticgroupSp(6,(cid:70) )andtheircohomologygroups
2
thus become Sp(6,(cid:70) )-representations. All computations are therefore
2
Sp(6,(cid:70) )-equivariant. We also study the mixed Hodge structures of these
2
cohomologygroups.
ThecomputationsforM [2]aremainlyviapointcountsoverfinitefields
3
whilethecomputationsforM [2]primarilyuseadescriptionduetoLooij-
3,1
enga [62] in terms of arrangements associated to root systems. This leads
ustothecomputationofthecohomologyofcomplementsoftoricarrange-
ments associated to root systems. These varieties come with an action of
thecorrespondingWeylgroupsandthecomputationsareequivariantwith
respecttothisaction.

Sammanfattning
Denna avhandling behandlar huvudsakligen kohomologin av modulirum-
menM [2]ochM [2]avkurvoravgenus3mednivå2-strukturutanre-
3 3,1
spektivemedenmarkeradpunktsamtnågranaturligadelrumavdessarum.
En icke hyperelliptisk kurva av genus 3 kan realiseras som en plan kvar-
tisk kurva och vi ägnar extra uppmärksamhet åt modulirummen Q[2] och
Q [2] av plana kvartiska kurvor utan respektive med en markerad punkt.
1
DensymplektiskagruppenSp(6,(cid:70) )verkarnaturligtpåderumsombetrak-
2
tasidennaavhandlingochkohomologigruppernaavdessarumblirsåledes
representationeravSp(6,(cid:70) ). Våraberäkningarärdärförekvivariantamed
2
avseende på gruppen Sp(6,(cid:70) ). Vi behandlar också blandade Hodgestruk-
2
turer.
BeräkningarnaförM [2]ärhuvudsakligenviapunkträkningaröveränd-
3
liga kroppar medan beräkningarna för M [2] främst använder en besk-
3,1
rivning av Looijenga [62] i termer av arrangemang associerade till rotsys-
tem.Dettaledertillberäkningaravkohomologinavkomplementtilltoriska
arrangemangassocieradetillrotsystem. RotsystemensWeylgrupperverkar
naturligtpådessavarieteterochvåraberäkningarärekvivariantamedavse-
endepådennaverkan.

Preface
Sinceitisabitunusual(atleastinSweden)towriteupathesisasamono-
graphratherthanacollectionofpapers,itmightbeappropriatetoexplain
thischoice.Firstandforemost,theresultspresentedhereareallquiteclosely
related and they are all building towards the same goal. Therefore, I felt
that they were most naturally presented in one cohesive story along with
therelevanttheory. Thereareofcoursedrawbacksofthischoice-themost
notableisprobablythattheresultingtextbecamebothlongertowriteand
read.However,“shortertoread”doesnotnecessarilymean“easiertounder-
stand”andthisformatalsoallowsavoidingtediousrepetitionsthatwould
occur in a collection where each paper itself would need to contain some
extentofbackgroundtheory.
Theintendedreaderofthisthesisisanalgebraicgeometer. Thismeans
thattheoryknowntomostalgebraicgeometersisusedwithoutmuchexpla-
nation,fortheorythatisknowntoexist(butperhapsnotknown)bymostal-
gebraicgeometersIprovidereferenceswhileIdiscusstheorymorespecific
to the topic in more detail. For example, the indended reader is expected
tobefamiliarwithsheavesandtoatleasthaveheardabouttheexistenceof
theconceptofmixedHodgestructureswhileAronholdsetsofthetacharac-
teristicsareexplainedinmoredepth.
Theexceptiontothisruleistheintroduction. Sinceatleastpartsofthe
topicathandareintherarebutfavorablepositionofbeingquiteelementary
we start at this point and the intended audience is therefore a bit broader
herethanintherestofthethesis. Attimesthischoiceleadstosimplifica-
tions,vaguenessandimprecisionbutIhopethisonlyoccursatpointswhich
areeitherwell-knowntoexpertsorcoveredinmoredetaillaterinthethesis.