Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications

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by  Aaron Michael Silberstein

ANABELIAN INTERSECTION THEORY I: THE CONJECTURE OF
BOGOMOLOV-POP AND APPLICATIONS
AARON MICHAEL SILBERSTEIN˚
3 January29,2013
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1. STATEMENT OF RESULTS
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A.Grothendieckfirstcoinedtheterm“anabeliangeometry”inalettertoG.Faltings[Gro97a]asaresponse
]
G toFaltings’ proofoftheMordell conjecture andinhiscelebrated Esquisse d’unProgramme [Gro97b]. The
A “yoga” of Grothendieck’s anabelian geometry is that if the étale fundamental group π´etpX,xq of a variety
1
. X at a geometric point x is rich enough, then it should encode much of the information about X as a
h
variety;suchvarietiesX arecalledanabelianinthesenseofGrothendieck,andhavethepropertythattwo
t
a
anabelianvarietieshaveisomorphicétalefundamentalgroupsifandonlyiftheyareisomorphic;andthatthe
m
isomorphismsbetweentheirétalefundamentalgroupsarepreciselytheisomorphismsbetweenthevarieties.
[
Grothendieck did not specify how much extra information should be encoded, and there is currently not a

2
consensusontheanswer. Ananabeliantheorem(orconjecture)isatheorem(orconjecture)whichasserts
v
8 thataclassofvarietiesareanabelian.
0 Grothendieck wrote in [Gro97a]about a number ofanabelian conjectures, one regarding the moduli of
6
curves,definedoverglobalfields(whichisstillopen);oneregarding hyperboliccurves,definedoverglobal
4
. fields;andabirationalanabelianconjecture, whichassertsthatSpecoffinitely-generated, infinitefieldsare
1
anabelian(inthiscase,wesaythefieldsthemselvesareanabelian). Theanabelian conjectureforhyperbolic
1
2 curves was proved in the 1990’s by A. Tamagawa and S. Mochizuki ([Tam97], [Moc99]). The birational
1
anabelianconjectureforfinitely-generated,infinitefieldsisavastgeneralizationofthepioneeringNeukirch-
:
v Ikeda-Uchida theorem forglobal fields ([Neu69], [Uch77], [Ike77], [Neu77]), and isnow atheorem due to
i
X F.Pop[Pop94].
r Grothendieck remarked that “the reason for [anabelian phenomena] seems...tolie in the extraordinary
a
rigidity of the full fundamental group, which in turn springs from the fact that the (outer) action of the
‘arithmetic’ partofthisgroup...isextraordinarily strong”[Gro97a].
F. Bogomolov had the surprising insight [Bog91] that as long as the dimension of a variety is ě 2,
anabelian phenomena can be exhibited — at least birationally — even over an algebraically closed field,
eveninthecompleteabsence ofthe“arithmetic”partofthegroupGrothendieck referenced.
Given a field K, we let G denote the absolute Galois group of K, the profinite group of field auto-
K
morphisms of its algebraic closure K (see [NSW08] for more details). Given two fields F and F , we let
1 2
IsomipF ,F q denote the set of isomorphisms between the pure inseparable closures of F and F , up to
1 2 1 2
Frobenius twists. Given two profinite groups Γ and Γ , we let IsomOutpΓ ,Γ q denote the set of equiva-
1 2 1 2
lenceclassesofcontinuous isomorphisms fromΓ toΓ ,moduloconjugation byelementsofΓ . Thereisa
1 2 2
canonical map
ϕ :IsomipF ,F q Ñ IsomOutpG ,G q (1)
F1,F2 1 2 F2 F1
˚UniversityofPennsylvania,Philadelphia,PA19104.Email:[email protected]
1
which,ingeneral, isneitherinjectivenorsurjective.
ThebirationaltheoryofavarietyofdimensionnoverK isencodedinitsfieldofrationalfunctions,and
everyfield finitely-generated overK andoftranscendence degree narises asthefield ofrational functions
ofaK-variety ofdimension n. F.Pop,developing Bogomolov’s insight, conjectured ananabelian theorem
for fields, finitely-generated and of transcendence degree n ě 2 over an algebraically closed field k. We
completetheproofof:
Theorem 1(TheConjecture ofBogomolov-Pop fork “ Q,F ). LetF and F befieldsfinitely-generated
p 1 2
and of transcendence degree ě 2 over k and k , respectively, where k is either Q or F , and k is
1 2 1 p 2
algebraically closed. Then ϕ is a bijection. Thus, function fields of varieties of dimension ě 2 over
F1,F2
algebraic closures ofprimefieldsareanabelian.
In [Pop11b], Popproved that if G » G then F and F have the same characteristic and transcen-
F1 F2 1 2
dence degree. Thus, the conjecture reduces to the case when F and F are of the same characteristic and
1 2
transcendence degree. Bogomolov and Tschinkel [BT08] provide a proof in the case of transcendence de-
gree“ 2whenk “ F . PopprovedthatϕisabijectionwhenF hastranscendence degreeě 2andk “ F
p 1 p
[Pop12];andwhenF hastranscendence degreeě 3andk “ Q[Pop11a]. Weprovethemissingcase:
1
Theorem 2 (The Birational Anabelian Theorem for Surfaces over Q). Let F and F be fields finitely-
1 2
generated andoftranscendence degree2overQ. Thenϕ isabijection.
F1,F2
The proof of Theorem2 is substantially different in structure from the other cases ofTheorem1. They
bothhavethesamestartingpoint, twotheoremsduetoPopfrom[Pop11a]anddescribed inSection3:
1. GivenasubgroupΓ Ď G thereisaprofinitegroup-theoretic recipe(Theorem25)whichdetermines
F
whetherornotΓisadecomposition orinertiagroupofaParshinchain(Definition 14).
2. GivenacollectionS “ tT uofinertiagroupsofrank-1Parshinchains(whicharegroup-theoretically
v
definablebyTheorem25),thereisarecipetodeterminewhetherthereisamodelX ofF —thatis,a
smoothvarietywithfunctionfieldF —forwhichS isthesetofinertiagroupsofWeilprimedivisors
centered onX. Inthiscase,S iscalledageometric setofprimedivisors(Definition 24).
Previous results took data such as these and reconstructed F directly, in a process which we now term
birational reconstruction. However, in our approach, we instead take the pair pG ,Sq and reconstruct a
F
model MpSq of F for which S is the collection of inertia subgroups of all prime divisors on MpSq. We
obtain a description of the geometry of MpSq without first reconstructing F, and we call this approach
geometricreconstruction. Themaintoolistheabilitytointerpretintersection theoryonMpSqusingonly
group theoretic recipes applied to S and G , without any knowledge of MpSq other than its existence;
F
this technique is the anabelian intersection theory of the title. In Section10 we show how Theorem2
follows from the geometric reconstruction results which take up the bulk of the paper. A generalization of
thegeometricreconstruction techniqueintranscendence degreeě 2forarbitrarycharacteristicwillcomein
asequeltothispaper.
Grothendieckalsoaskedfora“purelygeometric”descriptionofthegroupG .Anabeliangeometryover
Q
Qgivessuchadescription.1
Let X be an irreducible, geometrically integral, algebraic variety of dimension ě 2 defined over Q
so that X has no birational automorphisms defined over Q. We call such an X birationally, geometri-
cally rigid. Let k be the intersection of all subfields of Q over which a variety birationally equivalent
1Infact,Y.Iharaaskedamorerefinedquestion,whichT.OdaandM.Matsumotoraisedtoaconjecture:isG exactlytheouter
Q
automorphismsoftheétalefundamentalgroupfunctorfromthecategoryofQ-varietiestoprofinitegroups?Popshowed[Pop11b]
howbirationalanabeliantheoremsoverQcanbeusedtoprovideproofsofthisconjecture.Thus,Theorem2givesusanewproofof
theQuestionofIhara/ConjectureofOda-Matsumoto. Wewillelaborateonapplicationsofgeometricreconstructiontorefinements
oftheQuestionofIhara/ConjectureofOda-Matsumotoinalaterpaper.
2
to X is defined. Let QpXq be the rational function field of X. By our assumptions, QpXq is a field,
finitely generated overQ. Equation27showsthat G isageometric object: itistheinverse limitofthe
QpXq
profinitely-completed fundamental groups ofallcomplementsofQ-divisors inX.
Theorem3(TheGeometricDescription ofAbsoluteGaloisGroupsofNumberFields.). Thenatural map
G Ñ Out pG q (2)
k cont QpXq
is an isomorphism, and the compact-open topology induces the topology onG induced by its structure as
k
aGaloisgroup.
Theorem3answersGrothendieck’s question;thisisanalogoustoatheoremofI.BumaginandD.Wise,
whichrealizesanycountable groupastheouterautomorphism groupofafinitelygeneratedgroup[BW02].
InSection12,wewritedown,foranynumberfieldk,aninfinitefamilyofexplicitvarietiesX whichsatisfy
thehypotheses ofTheorem3.
Any group G which satisfies the hypotheses of Theorem3 is infinitely-generated and infinitely-
QpXq
presented. ItisthennaturaltoaskwhetherabsoluteGaloisgroupsofnumberfieldsareouterautomorphism
groupsoffinitely-generated, finitely-presented, profinite groupsofgeometricprovenance.
The “smallest group” currently considered to be a candidate to give such a geometric representation is
the group GT, whose study was initiated by Drinfel’d [Dri90] and Ihara [Iha91]. GT admits an injective
grouphomomorphism
y ρ: G Ñ GT. y (3)
Q
It is not known whether ρ is surjective. GT is a much-studied yet poorly-understood object; we review in
y
Section11thetheorywewillneed,andreferto[LS06]foramoreexhaustive survey.
WedenotebyM themodulispaceyofgenus0curveswith5distinct, marked,orderedpoints. Aspart
0,5
oftheconstruction ofGT,wehaveaninjection
y η :GT Ñ Outpπ1´etpM0,5qq, (4)
suchthatthecomposition y
η˝ρ:G Ñ Outpπ´etpM qq (5)
Q 1 0,5
istheinjection induced bythetheoryofthefundamental group.
WeproveanecessaryandsufficientcriterionforanelementofGT,asdeterminedbyitsimageunderη,
tobeintheimageofρ. M isbirationally equivalent toP2,whichgivesthescheme-theoretic inclusion
0,5
y
γ :SpecpQpx,yqq Ñ M (6)
0,5
of the generic point of M . The étale fundamental group functor then gives a continuous surjection of
0,5
profinite groups
γ : G Ñ π´etpM q,
˚ Qpx,yq 1 0,5
well-defined uptoconjugation byanelementofπ´etpM q.
1 0,5
Theorem4. Letα P imη.Thenα P imη˝ρifandonlyifαsatisfies thefollowing lifting condition: there
existsanautomorphism
α˜ : G Ñ G (7)
Qpx,yq Qpx,yq
sothatthediagram
3
G α˜ // G
Qpx,yq Qpx,yq
γ˚ γ˚
(cid:15)(cid:15) (cid:15)(cid:15)
π´etpM q α // π´etpM q
1 0,5 1 0,5
commutesuptoinnerautomorphisms.
ThisisthefirstnecessaryandsufficientgeometricconditionforanelementoftheGrothendieck-Teichmu¨ller
grouptolieintheimageofG .
Q
InSection12weprovide explicit examples of surfaces whichsatisfy thehypotheses ofTheorem3, and
weconclude withtheproofofTheorem4.
2. THE GEOMETRIC INTERPRETATION OF INERTIA AND DECOMPOSITION GROUPS OF
PARSHIN CHAINS
Definition 5. Let F be a field finitely generated over some algebraically closed field K of characteristic
zero; suchafieldwillbecalled afunctionfield. NotethatthefieldK isdetermined byF (forinstance, its
multiplicativegroupisthesetofalldivisibleelementsinthemultiplicativegroupofF);itwillbedenotedby
KpFq, andbecalled thefield ofconstants ofF. Thetranscendence degree ofF over KpFqwillbecalled
the dimension of F. Wewill denote by G the absolute Galois group of F and F the algebraic closure of
F
F.
For general theory of valuations, including proofs of the algebraic theorems cited without proofs, see
[FJ08]. Wewillalsouseresultsfrom[Ser56]and[Gro03]withimpunity.
Definition 6. A valuation v on F is an ordered group pvF,ďq, called the value group of v, along with a
surjective map
v :F Ñ vF Yt8u (8)
whichsatisfies
1. vpxq “ 8ifandonlyifx “ 0.
2. vpxyq “ vpxq`vpyq(here,wedefine8`g “ 8forallg P vF Yt8u.)
3. vpx`yqě mintvpxq,vpyqu,whereweextendtheorderingďtovF Yt8ubyg ď 8forallg P vF.
A valuation v gives rise to a valuation ring O , which is the set of all x P F such that vpxq ě 0. O is
v v
integrally closedinF andlocal,andwecallitsmaximalidealm . Wethendefinetheresiduefieldtobe
v
Fv “ O {m , (9)
v v
AsubringR Ď F isavaluation ringO forsomevaluation wifandonlyifforeveryx P Fˆ eitherx P R
w
orx´1 P R. Therefore, equivalently, wemaydefineavaluation v byitsplace
p : F Ñ FvYt8u (10)
v
whereO ismappedtoitsreductionmodm andFzO ismappedto8. AnymapfromafieldF toanother
v v v
fieldLwhichisaringhomomorphism “with8”thusgivesrisetoavaluation onF.
Definition7. Twovaluations willbecalledequivalentifandonlyiftheyhavethesamevaluation ring.
4
Definition8. Ifv isavaluationonF andwisavaluation onFv,thenwemaydefineavaluation w˝v on
F byconsidering thecomposition
p ˝p : F Ñ pFvqwYt8u (11)
w v
asaplacemaponF. Thisvaluation iscalledthecomposition ofwwithv.
Definition 9. A model X of a function field F is a smooth, connected KpFq-scheme of finite-type with a
map
s : SpecF Ñ X, (12)
X
thestructuremapofthemodel,whichidentifiesF withthefieldofrationalKpFq-functionsonX. Anormal
modelisamodelwiththerequirementofsmoothnessreplaced bynormality.
Definition10. Byvirtueofthestructuremap,themodelsformafullsubcategoryofthecategoryofschemes
under SpecF. We define an F-morphism of models to be a morphism of varieties under SpecF, and
BirpFqthefullsubcategory ofvarietiesunderSpecF whoseobjectsareprecisely themodelsofF.
Definition 11. Wesay that a valuation v on F has a center on or is centered on X if X admits an affine
open subset SpecA such that A Ă O , the valuation ring of v. Let R Ă F be a subring giving an affine
v
openSpecR ĎX. ThenthecenterofvonSpecRistheZariskiclosedsubsetofX definedasZpm XRq;
v
thecenterofv onX istheunionofthecentersofv onSpecRasSpecRrangesoverallaffineopensofX.
Wedenoteby|v|thecenterofv.
Definition12. Aprimedivisorv onF isadiscretevaluation trivialonKpFqsuchthat
tr. deg. Fv “ tr. deg. F ´1. (13)
KpFq KpFq
(This condition is very important in birational anabelian geometry in general, and we say that v has no
transcendencedefect;see[Pop94]formoredetails.)
Definition 13. A rank-1 Parshin chain for F is a prime divisor. A rank-i Parshin chain is a composite
w˝v,wherev isarank-pi´1qParshinchain, andwisaprimedivisor onFv.
Definition 14. We denote by Par pFq the collection of i-Parshin chains for F, where i ď tr. deg. F.
i KpFq
Given a rank-k Parshin chain v we denote by Par pvq the collection of rank-i Parshin chains of the form
i
w˝v. Par pvqisemptyifi ă k;istvuifi “ k;andisinfiniteifi ą k. IfS Ď Par pFqwedefine
i k
Par pSq “ Par pvq. (14)
i i
vPS
ď
Wealsolet
ParpSq“ Par pSqand ParpFq “ Par pFq. (15)
i i
i i
ď ď
Example15. Todescribetherank-1ParshinchainsonF,whereF isthefunctionfieldofasurfaceoverQ,
weconsider thesetofallpairspX,DqwhereX isapropermodelofF andDisaprimedivisoronX. On
thiscollection, wehaveanequivalence relationgenerated bytherelation „,wherewesay
pX,Dq „ pX1,D1q
ifthereexistsarational map
ϕ: X Ñ X1
5
respecting thestructure mapsofthemodelsX andX1 suchthatDismappedbirationally toD1 byϕ.
Therank-2Parshinchainsarethenequivalence classespX,D,pqwhereD isaprimedivisoronD and
pisasmoothpointonD,andtheequivalence relation „isnowgenerated bysaying
pX,D,pq „ pX1,D1,p1q
aslongasthereisarational map
ϕ: X Ñ X1
respecting thestructure mapsofthemodelsX andX1 sothatDismappedbirationally toD1 byϕandpis
mappedtop1 byϕ.
Givenanalgebraic extension L|F everyvaluation extendstoL,thoughnotnecessarily uniquely.
Definition16. WedefineX pL|FqtobethesetofvaluationsonLwhichrestricttovonF. IfL|F isGalois,
v
thenGalpL|Fqactstransitively onX pL|Fq. ForanyGaloisextensionL|F andv˜P X pL|Fqwedefinethe
v v
decomposition group
D pL|Fq “ tσ P GalpL|Fq |σpO q“ O u. (16)
v˜ def v˜ v˜
EachD pL|Fqhasanormalsubgroup, theinertiagroupT pL|Fq,definedasthesetofelementswhichact
v˜ v˜
astheidentity onLv˜.
Wehaveashort-exact sequence, thedecomposition-inertia exactsequence
1 Ñ T pL|Fq Ñ D pL|Fq Ñ GalpLv˜|Fvq Ñ 1. (17)
v˜ v˜
Ifv˜ ,v˜ P X pL|Fqandv˜ “ σv˜ forsomeσ P GalpL|Fq,then
1 2 v 1 2
D pL|Fq “ σ´1D pL|Fqσ andT pL|Fq “ σ´1T pL|Fqσ.
v˜1 v˜2 v˜1 v˜2
Thus,allD pL|FqandT pL|Fq,respectively,areconjugateforagivenv,andwhentheliftisnotimportant,
v˜ v˜
we denote some element of the conjugacy class of subgroups by D pL|Fq and T pL|Fq, respectively. We
v v
defineD andT ,respectively, tobeD pF|FqandT pF|Fq.
v v v v
ForanyGaloisextensionL|F,avaluationvonF andavaluationwonFv,wemaychoosev˜P X pL|Fq
v
andw˜ P X pLv˜|Fvq. Thereisthenanatural shortexactsequence, thecompositeinertiasequence:
w
1Ñ T pL|Fq Ñ T pL|Fq Ñ T pLv˜|Fvq Ñ 1, (18)
v˜ w˜˝v˜ w˜
whereT Ď G ,foranycomposite ofvaluations. Thus,ifT pL|Fqistrivial,
w Fv v˜
T » T .
w˜ w˜˝v˜
Wewillusethreedifferent typesoffundamental groups, withthefollowingnotation:
top
1. π will denote the topological fundamental group, the fundamental group of a fiber functor on the
1
top
category oftopological covers ofatopological space; covering space theory [Hat02]showsthat π
1
can be computed using based homotopy classes of maps into S1; this is the original fundamental
groupconsidered byPoincaré[Poi10].
2. πˆ will denote the profinite completion of πtop, the fundamental group of a fiber functor on the
1 1
category offinitetopological coversofatopological space.
3. π´et willdenotetheétalefundamental group.
1
6
For a normal variety X over an algebraically closed subfield of C, one has an equivalence between the
category of finite, étale covers of X and the finite, unramified covers of Xan, its corresponding analytic
spaceoverC,by[GR57]. Thisleadsimmediatelytothe
Theorem17(ComparisonTheorem). LetX beanormalvarietyoverCandx P XpCq.Thereisacanonical
isomorphism
πˆ pXan,xq» π´etpX,xq. (19)
1 1
Everyknown computation ofnonabelian fundamental groups ofvarieties factors through this compari-
son theorem; in characteristic p, for instance, this is combined with Grothendieck’s specialization theorem
[Gro03,X.2.4]toobtaininformation aboutfundamental groups.
Let now K “ C and F be a function field over C. Then we have the following interpretation of D
v
whenv isaprimedivisor. First,v isthevaluationassociated toaWeilprimedivisoronsomenormalmodel
X of F — that is, a normal variety with function field F, considered as a C-scheme. Given X, there is a
corresponding normalanalytic spaceXan. LetX beamodelofF onwhich|v|isaprimedivisor.
Example18. Theexceptional divisorE onBl pXq,theblowupatsomeclosedpointpofX,givesaprime
p
divisoronF butitscenteronX isnotcodimension 1andsoisnotcenteredasaprimedivisoronX.
Let D1 Ď X be the nonsingular locus of |v|; notice that the underlying topological space of D1 is
connected, asv isaprimedivisor.
Definition19. LetN thenbeanormaldiscbundlefor(equivalently, atubularneighborhood of)D1 and
T “ NzD1
thecomplementofD1 initsnormaldiscbundle N,whichadmitsthenormalbundlefibersequence
1 //F ι //T π // D1 // 1, (20)
whereF isoneofthefibers.
Let p be apoint on F. Note that F, like all fibers of π, isa once-punctured disk and thus is homotopy
equivalent toacircle. Thereisasurjection
ρ:G Ñ π´etpX,pq » πˆ pXan,pq (21)
F 1 1
(proof: each normal étale cover of X gives a normal extension of its field of functions) whose kernel we
willdefinetohavefixedfieldF ,andthefollowingcommutativediagram:
X
1 //πˆ pF,pq ι // πˆ pT,pq π // πˆ pD1,πppqq // 1 (22)
1 1 1
LLLLLLLLLL&& (cid:15)(cid:15)
π´etpX,pq.
1
Thenwehave
Proposition20(TheGeometricTheoryofDecompositionandInertiaGroups). Intheshortexactsequence
22:
1. Thetoprowisacentral extension ofgroups, asthenormalbundleiscomplex-oriented.
2. Theimageofπˆ pF,pqinπ´etpX,pqisaT pF |Fq.
1 1 v X
7
3. Theimageofπˆ pT,pqinπ´etpX,pqisaD pF |Fq.
1 1 v X
4. πˆ pD1,πppqqisaquotientofG ,corresponding tocovers ofD1 pulledbackfromcovers ofX.
1 Fv
Wewilldenote byt agenerator ofπtoppF,pq Ď πˆ pF,pq,aswellasanyofitsimagesinπ´etpX,pqor
v 1 1 1
T pF |Fq.
v X
Definition21. Werefertosuchat asameridianofv.
v
Each meridian is almost unique — its inverse also gives a meridian of v, albeit “in the opposite direc-
tion”. Thisshould beviewed asa“loop normal tooraround |v|”. Itsimagegenerates πˆ pF,pq. Ingeneral,
1
ifweareworkinginasituation inwhichwedonotspecify abasepoint, themeridianbecomesdefined only
uptoconjugacy.
Definition22. LetΓbeasubgroup ofagroupΠ. Thentheabelianization functorgivesamap
ab : Γab Ñ πab. (23)
1
WedenotebyΓa theimageofab. Inparticular, givenv avaluation, andΠaquotientofG orπtoppXqfor
F 1
some model of X, we will denote by Ta and Da the images of inertia and decomposition, respectively, in
v v
Πab,whichwillsometimesappearinthesequelasH . Weletta betheimageofameridianinΠab.
1 v
WecanalsodefinethemeridianofavaluationvonamodelX if|v|issmooth,andextendthedefinition
tonon-smooth |v|asfollows. Weresolvethesingularities of|v|onX togetabirational map
η : X˜ Ñ X
suchthat
1. η isanisomorphism outsideof|v|.
2. |v| Ď X˜ issmooth.
Thenwedefine ameridiant onX tobe
v
t “ η pt q (24)
v ˚ v
wheret isameridian onX˜. Toseethisiswell-defined, ifwehavetwosuchmapsη,η1 asinthefollowing
v
diagram:
X˜2 (25)
B
}
ϕ } B ϕ1
B
}
~~} B
X˜ X˜1
A
A |
AAη η1 ||
A |
A |
A |
A ~~||
X,
wemayalwaysconstructϕandϕ1 birational morphismssothat:
1. Theabovediagramcommutes.
2. ϕ,ϕ1,η˝ϕ,η1 ˝ϕ1 areisomorphisms outsideof|v|.
3. |v|issmoothinX˜2.
8
Inthiscase,themeridiansinX definedbyη1andηaretheimageofameridianinX˜2 underη˝ϕandη1˝ϕ1
so,bycommutativity ofthediagram, thetwomeridians arethesame.
There is also the notion of a meridian for a higher-rank Parshin chain; we give here the notion for a
rank-2Parshinchain.
Definition23. LetLbeanalgebraic extensionofF,w˝v arank-2ParshinchainonF,andX amodelof
F on which v has a center, but on which w ˝v is not centered. Then the meridian of the rank-2 Parshin
chain t for L|F is the element of AutpL|Fq induced by the inverse limit of the monodromies of a loop
w˝v
on |v| around the point induced by w on the normalization of |v|. As in Definition22, any image in an
abelianization willbedenotedta .
w˝v
LetF havedimension n. Thenifv isann´1-dimensional Parshinchain, Fv isthefunction fieldofa
curveoverK. Fv isequipped withafundamental, birational invariant: itsunramifiedgenusgpvq. Wecan
computethisasfollows:
gpvq “ rk Dva{xTva,Tay .
Zˆ p pPParnpvq
3. GEOMETRIC SETS AND THE MAXIMAL SMOOTH MODEL
Definition 24. We say that a set S of prime divisors of a function field F is a geometric set (of F) if and
only if there exists a normal model X of F such that S is precisely the set of valuations with centers Weil
primedivisors onX. Inthiscase,wewrite
S “ DpXq.
IfX issmooth,wesayX isamodelofS.
Theorem 25 (Pop). If F is a function field with KpFq “ Q, tr. deg. F ě 2, and let Γ Ď G be a
Q F
closed subgroup, up to conjugacy. Then there is a topological group-theoretic criterion, given one of the
representatives ofΓtodeterminewhetherthereexistsiandv P Par pFqsuchthatΓ “ T orΓ “ D ,and
i v v
whatthisiisifitexists.
This theorem is proven with G replaced by the pro-ℓ completion of G in [Pop11a]. To see this for
F F
G as a whole, we may apply Key Lemma 5.1 of [Pop11b]. As the maximal length of a Parshin chain is
F
thetranscendence degreeofF,thisrecipeimmediatelydeterminesthetranscendence degreeofF. Popalso
proved[Pop11b]:
Theorem26(Pop). GivenageometricsetS ofprimedivisors onF,
1. IfS isageometricsetofprimedivisorsonF,thena(possiblydifferent)setS1 ofprimedivisorsonF
isageometricsetifandonlyifithasfinitesymmetricdifference withS.
2. Thereexistsagroup-theoretic recipetorecover
GeompFq “ ttpT ,D q |v P Su| S ageometricsetu.
v v
Definition27. IfS isanysetofprimedivisors onF,wedefinethefundamentalgroupofS tobe:
ΠS “ GF{xTvyvPS.
Here,xTvyvPS isthesmallestclosed, normalsubgroup ofGF whichcontains everyelementinevery conju-
gacyclassineachT . IfT Ď S isasubset, thenwedenoteby
v
ρTS :ΠT Ñ ΠS
therestriction map,anddropsubscripts whentheyareunambiguous.
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Given ageometric setS, there are many possible X such that DpXq “ S; forinstance, any model less
afinitesubsetofpointshasthesamesetofprimedivisors. Wenowdefinethemaximalmodelonwhichwe
willbeabletoeffectourintersection theory.
Theorem 28. LetS be a geometric set for a function field F of dimension 2. There exists a unique model
MpSqofF suchthatthefollowing holds:
1. DpMpSqq “ S.
2. MpSqissmooth.
3. πˆ1pMan,pq» π1´etpMpSq,pq » ΠS.
4. If X is any other smooth model of F which satisfies S “ DpXq, then there exists a unique F-
morphism X Ñ MpSq,andthisisasmoothembedding.
Proof. LetU beamodelofS,andletX beasmoothcompactification ofU. Let
B “ DpXqzDpUq
be the collection of field-theoretic prime divisors in the boundary of U in X. This is a finite set, as the
boundary divisor is itself a finite union of prime divisors. We now define a sequence of pairs pX ,B q of
i i
varietiesX andfinitesetsofdivisors B Ă DpXqinductively asfollows:
i i
1. LetX “ X,B “ B.
1 1
2. Wenow construct pX ,B qfrom pX ,B q. First, take thecollection tv u Ď B suchthat each|v |
i`1 i`1 i i j i j
isa´1-curve such that noother |v1|that intersects itinthe boundary is a´1-curve, and blow down.
SetX tobethisblowdown,andB “ B ztv u.
i`1 i`1 i j
AsB isfinite,atsomepoint, thissequence becomesstationary—let’ssayatpX ,B q. Thenwedefine
1 n n
U “ X z |v|. (26)
max n
vďPBn
Toprovethatthissatisfies property 4,letU1 beanothermodelandX1 asmoothcompactification ofit. Run
the algorithm on X1 to get a pair pX1 ,B1 qThen by strong factorization for surfaces (see Corollary 1-8-4,
n1 n1
[Mat02]),thereexistsaroof
Y
B
} B
} B
~~}}δ}}}} BBBδB1B
X1 X ,
n1 n
whereδ andδ1 arebothsequences ofblowups. Foranymorphism
ϕ : Z Ñ Z1
ofvarieties wemaydefinetheexceptional locus
Epϕq “ p P Z1 |dimpϕ´1ppqq ě 1
Letp P Epδ1qXU . Thenδ1´1ppqiscon nected, soδpδ1´1ppqqisalso(connected. Itisproper, soiseither
max
aunionofdivisorsorapoint. Ifitisaunionofdivisors andoneofthesedivisorswereinU1 ,thisdivisor
max
10