Anabelian Intersection Theory

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Anabelian Intersection Theory

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Citation Silberstein, Aaron. 2012. Anabelian Intersection Theory. Doctoral dissertation, Harvard University.

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:10086302

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Dissertation Advisor: Professor Florian Pop Aaron Michael Silberstein

Anabelian Intersection Theory

Abstract

Let F be a field finitely generated and of transcendence degree 2 over _Q. We describe a correspondence between the smooth algebraic surfaces X defined over _Qwith field of rational functionsF and Florian Pop’s geometric sets of prime divisors onGal(F /F),which are purely group-theoretical objects. This allows us to give a strong anabelian theorem for these surfaces. As a corollary, for each number fieldK, we give a method to construct infinitely many profinite groupsΓsuch thatOutcont(Γ)is isomorphic toGal(K/K),and we find a host of new categories

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Acknowledgements

The embryo of the idea for this thesis was conceived at a Trivial Notions talk by Anand Patel, developed limbs on the streets of Hồ Chí Minh City, and gestated at the Hausdorff Institute in Bonn, the Băleanu Institute for Advanced Studies in Bucures,ti, and the Suboti´c Center for

Mathematics in Beograd. Its birth was a long and painful process, which culminated at Harvard and in Turkey. I thank my hosts for their hospitality during my visits, and recognize that this work might not have arrived without the stimulating environment they provided.

I was supported financially by Harvard University, an NSF GRFP stipend, and an ASEE NDSEG fellowship.

Florian Pop explained to me his theory of geometric sets until I could see mold it to “see geometry,” and encouraged this project. Ethan Street helped me translate my intuition for non-transverse intersections into language topologists can understand, and David Massey showed me why Reeves’ lemma is obvious. Dialogue about valuations with Adam Topaz helped immensely while I was developing this theory. I have also benefited from discussions with Sinan ¨Unver, David Geraghty, Glenn Stevens, K¨urs¸at Aker, Matt Emerton, Chris Skinner, Adrian Iovit,ă, Henri

Darmon, Tamás Szamuely, Jon Hanke, Kirsten Wickelgren, and Alex Suciu.

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Koberda, Antoni Rangachev, Andrei Negut,, S,tefania Bercu, Cristina Popa, Alex Petrescu, Tamara

Băleanu, George Zamfir, Simion Filip, Corina Panda, and Sorina Vasile, and being on the road were some of my best memories from this period of my life. Mathematical life in the time I spent in Bucures,ti was lively because of Alex Popa, Vicent,iu Pas,ol, Nicu Beli, and Adrian Diaconu.

I am especially thankful to Grandma Eva and Grandpa Fred; the Napa Bronks; the Woodside Bronks; the Mirmonsef-Takloo-Bighash family; the Da˘gdelen family; the Băleanu family; the Bercu family; the Petrescu family; the Subotić family; the Proistosescu family; the Păcurar family; the Papaioannou family; Đỗ Hoàng Kim; the Körner family; the Grin-Tarlow family; Andreea Nicoara; the Cviˇcek family, and its Los Angelesian extension; the Panda family; the Negoescu family; and the Kamburov family for making me feel at home in so many places around the globe. I have been lucky to have the Boston Bronks — Pete, Susan, Gabe and Johanna - nearby. I entered graduate school with Thomas Koberda, Jack Huizenga, Katy Körner, Tanya Kobylyatskaya, Ivana Bozić, Ana Caraiani, Jonathan Barlev, Ethan Street, Daniel Kane, Yi Li, and Yu-Shen Lin, and I could not have asked for a better cohort. Jam sessions with Adnan, cooking Ethiopian food with Andreea, HONK! with Corina, long walks with Veronika in Cambridge and Sorina in Bucures,ti,

Asia and Chevalley-Weil with Thomas, Kyoto with Adam, Andrew, Ascher, Sarah, and Chris, and duck parties with my cohort remain highlights of graduate school. Esmé Wang has been a constant font of support. To anyone else not mentioned here, but who played a part in my life during graduate school: thank you!

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from Curt McMullen. If I omitted your name here, please forgive me.

The students I have been able to work with while at Harvard — especially Krishna Dasaratha, Lucia Mocz, Eva Belmont, Michael Jemison, and John Sheridan — kept me on my toes.

Barry Mazur and Sophie Morel read my thesis and their comments and attention proved invaluable in ensuring its legibility and correctness.

Florian Pop has been both the single most important mathematical influence on my life and a dear friend, and I am lucky to have met him.

Dick Gross made sure I made certain correct choices at Harvard, and I am continually grateful for his wisdom.

Mark Goresky has provided support at every step of my career, and his mathematical influ-ence on me cannot be underestimated.

I am in mathematics because of Glenn Stevens, and he has been a fantastic mentor and a true friend since I was first interested in mathematics. I learned much of the basic mathematics I know from David Fried and Steve Rosenberg. The friends I made through PROMYS — especially Dustin Clausen, Dan Jerison, Agnès Beaudry, Claudia Scheimbauer, Ila Varma, Kevin Hughes, David Coley, Jon Hanke, Ander Steele, Big Papa, Maya Vinokour, Cameron Freer, Lucas Culler, Seo Hyung Kim, and Jay Pottharst — have a very special place in my heart. Jay Pottharst and Cameron Freer, especially, have been of great mathematical and moral support to me while this thesis was being written.

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Index of Notation

>, 55

a_{, 21}

A(S), 45

a(S), 45

Bir(F), 15

∂S0(S), 45

|D|, 57

∆, 32

∆s_(p)_{, 58}

dim, 59

Div(S), 55

D−, 55

D+, 55

Dv, 17

D˜v, 17

´

Et(S), 65

Fam(F), 57

F, 13

Fv, 14

g(B), 48, 49

GBir(F), 66

GBirmax(F), 54

Geom(F), 23

GF, 13

g(v), 22

H, 47, 49

IB, 48, 49

(D1·D2), 33 i(p, C1·C2;X), 28 (v1·v2), 56

ι, 42

Lim, 44–45

mp(v), 56 M(S), 24

µ, 56

mv, 14 Ov, 14

P, 32

Par,Pari, 16

Pgeom_{, 54}

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PS0(S), 45

pv, 14

(p, v·w;S), 43

ρT S, 24

supp(D), 55, 60

sX, 15

Tv, 17

tv, 20

ta v, 21

T˜v, 17

tw◦v, 22

ta w◦v, 22

V, 47, 49

v, 14

vF, 13 |v|, 15

Xan_{, 19}

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Index of Definitions

Absolutely uncentered point, 45 Adequate divisor, 61

Algebraic equivalence, group-theoretical, 57

Base locus, 60 Base point, 60 Basepoint free, 60 Boundary points, 45

Candidate points, 45 Center, 15

Complete family, 57

Complete linear system, group-theoretical, 57

Decomposition group, 17 Dimension, 13

in projective geometry, 59 Divisor

effective, 55 Divisor group, 55 Divisor, prime, 15 Domination, 64

Equivalence, 65

Étale site, of a geometric set, 65

F-morphism, 15 Field of constants, 13 Function field, 13 Fundamental group, 65

Fundamental group of a geometric set, 23

Geometric points, 54

Geometric reconstruction, 62 Geometric set, 23

proper, 54

Horizontal-vertical Decomposition, 47, 49

Inertia group, 17 Interior points, 45 Intersection, 32

Intersection pairing, 56

Limits, 44–45

Linear closure, in projective geometry, 59 Linear equivalence, group-theoretical, 57 Linear independence, 59

Local intersection number, 28, 43

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Maximal smooth, 24 Normal, 15

Multiplicity, 56

Noncuspidal chain, 44 Nonnodal chain, 44

Numerical equivalence, group-theoretical, 57

Parshin chain, 15 Place, 14

Point, 32

Projectivized jet space, 59 Projectivized tangent space, 59 Projectivizing datum, 61

Recognition, 32 Residue field, 14

Separates points, 60 Separates tangent lines, 60 Smooth chain, 44

Structure map, 15 Support, 55

Support of a set of divisors, 60

Tangent, 58

Total intersection product, 33 Transcendence defect, 15

Unramified genus, 22

Valuation, 13

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Chapter 1 Introduction

1.1 Grothendieck, Torelli, and Schottky

Since the dawn of what we now know as algebraic geometry, it has been recognized that there are two types of fundamental invariants of algebraic varieties: “discrete” invariants, and “continuous” invariants. Implicit in the work of Riemann is a phenomenon that has dictated the shape of algebraic geometry:

1. Discrete invariants are topological.

2. Continuous invariants are geometric, analytic, or algebraic.

These discrete invariants are usually given by the machinery of homology/cohomology and fun-damental groups. The continuous invariants are then realized asstructureson these objects; one of the most productive, for instance, is a Hodge structure.

Nowhere is this more evident than in the theory of projective algebraic smooth curvesC over the complex numbers_C:

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topological surface. There is a single “discrete invariant” of a smooth, projective curveC: a positive integer, thegenusof the underlying compact, topological surfaceC(_C) :

gC =

1

2dimH1(C(C),R).

IfC1, C2are two algebraic curves with the same genus,

(a) For any two pointsp1 ∈ C1(C), p2 ∈ C2(C), π1(C1, p) ' π1(C2, p0);in fact, C1(C)

andC2(_C)are Eilenberg-Mac Lane spaces, and are diffeomorphic.

(b) There exists a smooth, but not necessarily projective, algebraic curveC0, two points

p1, p2 onC0, and a smooth, algebraic varietyX with a morphismϕ : X → C0 such

that

i. ϕ−1(p1)'C1 andϕ−1(p2)'C2.

ii. ϕis, topologically, a fibration.

Here, the “discrete invariant” traditionally understood as a number is fundamentally used in its guise as a discrete group.

2. To determine a specific algebraic curve from all the algebraic curves sharing its genus (and thus fundamental group), we only need to look at the Hodge structure on the Lie algebra of the maximal2-step nilpotent quotient of its topological fundamental group; this is the classical Torelli theorem [And58]. Classically, this would be said as: the periods of an algebraic curve — a set of complex numbers, which can vary continuously — determine the algebraic curve.

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This gives rise to two general classes of problems. The first, which we call theTorelli problem, is:

Problem 1. Given two algebraic varietiesX1, X2 with a given set of “discrete invariants,” how

much structure do we need to add to these invariants in order to determine whetherX1 'X2?

On the other hand, consider the problem of Schottky:

Problem 2. Given a class of algebraic varieties{Xi}which share a topological invariantΓ, how do we know when an algebraically induced structure onΓcomes from someXi?

These are two central questions in algebraic geometry, and I do not believe there currently exists a satisfactorily general and precise formulation of either of them.

In hisEsquisse d’un Programme[Gro97b] and letter to Faltings [Gro97a], Grothendieck put forward a program for translating questions in arithmetic and algebraic geometry into questions about profinite groups viaπ´₁et, the étale fundamental group.

In general, letX be a geometrically integral, algebraicK-variety,X_K its base-change toK, the algebraic closure ofK, andxa geometric point ofX. Thenπ´et₁ sits in a canonical short exact sequence

1→π₁´et(X_K, x)→π₁et´(X, x)→π´et₁ (SpecK, x)→1.

Sinceπ´et

1 (SpecK, x) ' GK, the absolute Galois group ofK, conjugation by the quotient gives an outer action ofGKonπ´et1 (XK, x).

The “yoga” of Grothendieck’s anabelian geometry is that if the fundamental groupπ´et 1 (X)is

“rich enough”, then it should encode much of the information aboutXas a variety; such varieties

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1. What kind of information aboutXis captured byπ´et₁ (X, x)and, in particular, whichXare anabelian?

2. What group-theoretic properties characterize the étale fundamental groups among all profi-nite groups?

3. Which maps between étale fundamental groupsπ´et

1 (X, x)→π1´et(X0, x0)come from maps

of their underlying varieties?

Indeed, Grothendieck was inspired by arithmetic; Hodge theory can be mimicked through Galois actions, and he thought that an étale fundamental group being “rich enough” involved an arithmetic Galois action. In short, fundamental groups — broadly interpreted — are com-plicated/interesting enough in arithmetic settings that they can incorporate both topological and continuous invariants. In fact, proving a variety is anabelian proves that it satisfies a generalized Torelli problem. The second of the above questions is a strong Schottky problem; there is very little progress on this problem, however.

1.2 Reconstruction Techniques for Fields: Valuations, and

Ga-lois Theory

As we will use it, areconstruction techniqueis an algorithm for recovering an algebro-geometric object from its Galois or fundamental group by using only group-theoretic recipes on that group. The goal of this thesis is to describe a new reconstruction technique for function fields of surfaces, which makes fundamental use of classical algebraic geometry.

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de-composition and inertia groups of valuations on number fields from their absolute Galois groups [Neu69]. This was then developed in [Uch77] and [Ike77] (and Iwasawa, in unpublished work) to give the following theorem (which is also exposited in [Neu77]):

Theorem 3. Let K1, K2 be number fields. Then Isom(K2, K1) is in natural bijection with

IsomOut(GK1, GK2), the quotient of the isomorphism set between their absolute Galois groups

by conjugation byGK2.

There is a corresponding theorem for global fields of positive characteristic, but it is slightly more complicated and we will not be dealing with the effects of positive characteristic at all in this thesis. Pop [Pop94, Sza04], finally, extended these techniques to show

Theorem 4. LetK1, K2 be finitely generated fields of characteristic zero. ThenIsom(K2, K1)

is in natural bijection withIsomOut(GK1, GK2),the quotient of the isomorphism set between the

two groups by conjugation byGK2.

In the early 1990s, Bogomolov suggested that birational anabelian phenomena should mani-fest also for higher-dimensional function fields over algebraically closed basefields, thus in total absence of arithmetic Galois actions [Bog91]. The strongest evidence of this is Bogomolov’s idea that commuting liftable pairs in Galois groups of certain fields originate from a special class of valuations. The ideas in valuation theory are developed in [Bog91], [BT11], [Pop10], and [Pop11a]. We will take the recent preprint of Topaz [Top12] as the definitive reference, as it encompasses, generalizes, and unifies all previous work on the valuation-theoretic side of this program.

Let now F be a field, finitely-generated over _Q, and of transcendence degree 2. We will construct a natural equivalence of categories

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whereBir(F)is the category whose objects are smooth varieties with function fieldF and whose morphisms are birational maps between such varieties; andGBir(F)is a category whose objects are constructed and whose morphisms are defined by purely group-theoretical recipes fromGF, the absolute Galois group of F. We do not merely give the category GBir(F); from the group theory, given an objectXofGBir(F)and its corresponding modelX =M(X), we have explicit recipes to:

1. Determine ifX is proper.

2. Compute Betti numbers forXand ifXis proper, compute Hodge numbers.

3. ComputeNS(X),the Néron-Severi group ofX, ifX is proper.

4. Compute the closed points ofX.

5. Identify the prime divisors on X and compute local and global intersection numbers of divisors.

6. Calculate global sections of line bundles associated to divisors onX ifX is proper.

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(Kronecker dimension 1), and the induction step on Kronecker dimension uses the theory of relative curves.

On the other hand, in the geometric situation, the reduction to curves is not possible, as the absolute Galois group of the function field of a curve over an algebraically closed field of characteristic zero depends only on the cardinality of the basefield, and in particular does not carry any information — geometric or otherwise — about the curve. The strategy used in, for instance [Pop10], is to recover the function field directly from its absolute Galois group, by first recovering the group of principal divisors of the field, and then interpreting this as the projectivization of the additive group of the function field. Once we have this, we may deduce, by using the fundamental theorem of projective geometry, the full field structure.

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varieties, which are not necessarily anabelian.

The second chapter of the thesis reviews the work already done to complete Bogomolov and Pop’s program to recover valuations for _Q. We do not give proofs or develop the theory of commuting liftable pairs as in [Pop11b] and [Top12]; rather, we take these procedures as a black box for the rest of the reconstruction. In particular, given the function fieldF of a smooth surface

X defined over_Q, we need to define a certain class of valuations onF — theParshin chains— which have geometric meaning. The inspiration — the classical (partial) Parshin chains — are defined on a varietyXover an algebraically closed fieldK, and are defined as chains

X =D0 ⊃D1 ⊃D2 ⊃D3· · · ⊃. . . Dn

wheren ≤dimX, and thisnis called therankof the chain. EachDi is a divisor onDi−1. The

valuations we are interested in are birational versions of the classical Parshin chains. These are Parshin chains on a variety birationally equivalent to X. This definition has, then a completely field-theoretic description in terms of valuations. A Weil prime divisor on a smooth variety birationally equivalent to X gives a valuationv on K(X), discrete, trivial on K, and with no transcendence defect: the transcendence degree of the residue field of v overK is one less than the transcendence degree ofK(X)overK. Such a valuation is called aprime divisorofK(X). Conversely, for each prime divisor of K(X) there is a Weil prime divisor on a smooth variety birationally equivalent to X. We define the rank one birational Parshin chains to be the prime divisors onK(X). The birational notion of a prime divisorv2“on a prime divisorv1” — to make

longer Parshin chains — will be a prime divisorv2on the residue field ofv1. A birational Parshin

chain will be defined to be a composite vn ◦vn−1 ◦ · · · ◦v1 of valuations, where vi is a prime divisor on the residue field ofvi−1. Since we work exclusively with birational Parshin chains, we

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chain.

Each valuation onK(X)has a decomposition and inertia group in the absolute Galois group

GK(X)ofK(X). In the case of surfaces, we describe the geometric and algebraic interpretations

of the decomposition and inertia groups corresponding to a Parshin chain. We then recall that there is a recipe to determine, purely from the group theory ofGK(X), whether a given subgroup

is the decomposition or inertia group of a Parshin chain.

For any model X0 of the function fieldK(X), the corresponding set SX0 of all the prime divisors onK(X)(that is, considered as valuations), whose centers are codimension1onX0, is called the geometric setof prime divisor defined byX0. In general, we say that a set of prime divisors is a geometric set if it is equal toSX0 for some smooth modelX0. We recall that there is a group-theoretic algorithm to determine whether a set of subgroups is exactly the set of inertia groups corresponding to the elements of a geometric set for some modelX0 ofK(X).

The third chapter concerns the reconstruction procedure itself, which goes in roughly four steps:

1. LetD1, D2 be divisors on a smooth surfaceX over_Q with function fieldF, and letpbe a point on D1. Hence, D2 determines a rank-1 Parshin chain onF; and the pair(D1, p)

determines a rank-2Parshin chain on F. We show how to compute agroup-theoretic lo-cal intersection number of D1 and D2 at pusing the inertia groups of the Parshin chains

corresponding to D2 and (D1, p) and the geometric set SX as structures on GF, which agrees with the geometrically defined local intersection number (D1, D2)p as defined in, for instance, [Ful98], subject to certain global geometric conditions.

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on the interior of the maximal smooth model.

3. We then use this intersection theory to detect when the maximal smooth model of a geo-metric set is a fibration of curves by curves.

To see one reason this is subtle, consider the following question, to which I suspect a positive answer, but which I cannot produce: LetK be a countable, algebraically closed field of characteristic zero. LetX be a quasi-projective, algebraic surface overK (hence reduced, separated, etc.). Let{Ci}i∈_Nbe a set of prime divisors ofX, such that:

(a) S

i∈NCi(K) =X(K).

(b) Ci(K)∩Cj(K) = ∅fori6=j.

(c) All Ci have the same genus and number of punctures (that is, are in the same con-nected component of the moduli of curves).

Do theseCi then form an algebraic family withinX?

4. We then can finish the reconstruction, in two different ways. We take a geometric set whose maximal smooth model is proper, determine which rank-1Parshin chains on it correspond to very ample divisors, and we use the Fundamental Theorem of Projective Geometry, as in [Art88], to reconstruct the projective coordinate ring of that maximal smooth model group-theoretically. On the other hand, we can cover the variety with fibrations of curves over curves as in the last step, in order to construct a topological model — and in fact, to reconstruct its étale site — as a space glued out of open sets that areK(π,1)’s.

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Theorem 5(The birational anabelian theorem for surfaces over_Q). LetF1andF2 be two fields, finitely-generated and of transcendence degree2over_Q. Then the natural map

ϕ: Isom(F2, F1)→IsomOutcont(GF1, GF2)

is a bijection (whereIsomdenotes all field-theoretic isomorphisms, and not merely the_Q-isomorphisms); and in particular,F1 andF2 are isomorphic as fields if and only if their absolute Galois groups

are isomorphic as topological groups.

This result goes beyond the (unpublished) [Pop11a], where the Birational Anabelian Theorem is proven for transcendence degree≥3. Our result can also be used as the base case in an induc-tion procedure to reprove the Birainduc-tional Anabelian Theorem for higher-dimensional varieties; we will not do that in this thesis, however.

Recall that a varietyXover a fieldkisbirationally rigidifk(X)has no nontrivial field auto-morphisms, andgeometrically, birationally rigidif there are no nontrivialk-automorphisms of

k(X). Note that ifXis birationally rigid and geometrically, birationally rigid, thenAut(k(X)) = Gk. We have then as an immediate corollary of Theorem 5.

Theorem 6. Letkbe a number field. LetXbe a birationally rigid and geometrically, birationally

rigid surface overk. ThenOutcont(Gk(X))'Gk.

The (short) fourth chapter involves writing down explicitly surfaces which satisfy the hy-potheses of Theorem 6.

The conjecture of Ihara/Oda-Matsumoto is:

Conjecture 7. LetV/k be the category of geometrically-based varieties over a number field k. Letπ₁´etbe the étale fundamental group functor

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taking values in the category of profinite groups. Then

Gk 'Out(π´1et),

whereOutis the automorphisms of the functor modulo the automorphisms gotten by compatible

systems of inner automorphisms.

For a proof of Conjecture 7 and its history, see [Pop11b]. We may deduce, by replacing the partial birational reconstruction in [Pop11b] with Theorem 5, a refinement of Conjecture 7 (which, by [Pop11b] proves Conjecture 7):

Theorem 8. LetX be a birationally rigid and geometrically, birationally rigid surface over k,

and letV be any full subcategory ofVkso that for any open subvarietyU ⊆X, there is an open subsetU0 ⊆U ⊆Xsuch thatU0 is an object ofV. Then

Gk 'Out(π´et1 |V).

As a corollary of geometric reconstruction, we may obtain a result in this direction which we cannot access using birational reconstruction.

Theorem 9. LetV be the full category containing all the open subsets ofM0,5, the moduli space

of5points inP1, as in [HS00], and enough mapsϕ:M0,5 → M0,4to kill all automorphisms of

the full subcategory ofV containingM0,5 andM0,4. Then

G_Q 'Out(π´₁et|V).

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Chapter 2 An Anabelian Cookbook

2.1 The Geometric Interpretation of Inertia and

Decomposi-tion Groups of Parshin Chains

Definition 10. LetF be a field finitely generated over some algebraically closed fieldK of

char-acteristic zero; such a field will be called a function field. Note that the field K is determined

byF (for instance, its multiplicative group is the set of all divisible elements in the multiplicative

group of F); it will be denoted by K(F), and be called thefield of constants of F. The

tran-scendence degree ofF overK(F)will be called thedimensionofF. We will denote byGF the

absolute Galois group ofF andF the algebraic closure ofF.

For general theory of valuations, including proofs of the algebraic theorems cited without proofs, see [FJ08]. We will also use results from [Ser56] and [Gro03] with impunity.

Definition 11. A valuation v on F is an ordered group(vF,≤), called thevalue group of v,

along with a surjective map

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which satisfies

1. v(x) = ∞if and only ifx= 0.

2. v(xy) = v(x) +v(y)(here, we define∞+g =∞for allg ∈vF ∪ {∞}.)

3. v(x+y)≥min{v(x), v(y)},where we extend the ordering≤tovF∪ {∞}byg ≤ ∞for allg ∈vF.

A valuationvgives rise to avaluation ringOv, which is the set of allx∈ F such thatv(x)≥0. Ov is integrally closed in F and local, and we call its maximal ideal mv. We then define the

residue fieldto be

F v =Ov/mv, (2.2)

A subringR ⊆ F is a valuation ringOw for some valuationwif and only if for every x ∈ F× eitherx∈Rorx−1 ∈R. Therefore, equivalently, we may define a valuationvby itsplace

pv :F →F v∪ {∞} (2.3)

whereOvis mapped to its reduction modmv andF \ Ovis mapped to∞. Any map from a field

F to another fieldLwhich is a ring homomorphism “with ∞” thus gives rise to a valuation on

F.

Definition 12. Two valuations will be calledequivalentif and only if they have the same valuation

ring.

Definition 13. v is a valuation onF andwis a valuation onF v, then we may define a valuation

w◦v onF by considering the composition

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as a place map onF. This valuation is called thecompositionofwwithv.

Definition 14. AmodelX of a function fieldF is a smooth, connectedK(F)-scheme of

finite-type with a map

sX : SpecF →X, (2.5)

thestructure map of the model, which identifiesF with the field of rationalK(F)-functions on

X. Anormal modelis a model with the requirement of smoothness replaced by normality.

Definition 15. By virtue of the structure map, the models form a full subcategory of the category

of schemes under SpecF. We define anF-morphism of models to be a morphism of varieties

under SpecF, and Bir(F) the full subcategory of varieties under SpecF whose objects are precisely the models ofF.

Definition 16. We say that a valuationv onF has a center oris centeredonX ifX admits an

affine open subsetSpecAsuch that A ⊂ Ov, the valuation ring ofv. LetR ⊂ F be a subring giving an affine open SpecR ⊆X. Then thecenterofv onSpecRis the Zariski closed subset ofX defined asZ(mv ∩R);the center ofv onX is the union of the centers ofv onSpecR as

SpecRranges over all affine opens ofX. We denote by|v|the center ofv.

Definition 17. Aprime divisorv onF is a discrete valuation trivial onK(F)such that

tr. deg._K₍_F₎F v = tr. deg._K₍_F₎F −1. (2.6)

(This condition is very important in birational anabelian geometry in general, and we say thatv

hasno transcendence defect; see [Pop94] for more details.)

Definition 18. A rank-1 Parshin chainfor F is a prime divisor. A rank-n Parshin chain is a

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Definition 19. We denote by Pari(F)the collection of i-Parshin chains forF. Given a rank-k

Parshin chainv we denote by Pari(v)the collection of rank-iParshin chains of the formw◦v.

Pari(v)is empty ifi≤k; is{v}ifi=k; and is infinite ifi≥k. IfS ⊆Pari(F)we define

Pari(S) =

[

v∈S

Pari(v). (2.7)

We also let

Par(S) =[

i

Pari(S)and Par(F) =

[

i

Pari(F). (2.8)

Example 20. To describe the rank-1 Parshin chains on F, where F is the function field of a

surface over_Q, we consider the set of all pairs(X, D)where X is apropermodel ofF andD

is a prime divisor on X. On this collection, we have an equivalence relation generated by the

relation

(X, D)∼(X0, D0)

if there exists a dominant map

ϕ:X →X0

respecting the structure maps of the models XandX0 such thatDis mapped birationally toD0

byϕ.

The rank-2Parshin chains are then equivalence classes(X, D, p)whereDis a prime divisor

onDandpis a smooth point onD, and the equivalence relation is now generated by

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if

ϕ:X →Y

so thatDis mapped birationally toD0 byϕandpis mapped top0 byϕ.

Given an algebraic extensionL|F every valuation extends toL, though not necessarily uniquely.

Definition 21. We define Xv(L|F)to be the set of valuations onL which restrict tov on F. If

L|F is Galois, thenGal(L|F)acts transitively on Xv(L|F). For any Galois extensionL|F and

˜

v ∈ Xv(L|F)we define thedecomposition group

Dv˜(L|F) = {σ ∈Gal(L|F)|σ(O˜v) =O˜v}. (2.9)

EachDv˜(L|F)has a normal subgroup, theinertia groupT˜v(L|F), defined as the set of elements which act as the identity onL˜v.

We have a short-exact sequence, thedecomposition-inertia exact sequence

1→Tv˜(L|F)→Dv˜(L|F)→Gal(L˜v|F v)→1. (2.10)

If˜v1,v2˜ ∈ Xv(L|F)andv1˜ =σv2˜ for someσ ∈Gal(L|F), then

Dv˜1(L|F) =σ

−1_D ˜

v2(L|F)σandT˜v1(L|F) = σ

−1_T ˜

v2(L|F)σ.

Thus, all D˜v(L|F)and Tv˜(L|F), respectively, are conjugate for a given v, and when the lift is

not important, we denote some element of the conjugacy class of subgroups by Dv(L|F) and

Tv(L|F), respectively. We defineDv andTv, respectively, to beDv(F|F)andTv(F|F).

For any Galois extensionL|F, a valuationv onF and a valuationwonF v, we may choose

˜

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inertia sequence:

1→Tv˜(L|F)→Tw◦˜ ˜v(L|F)→Tw˜(L˜v|F v)→1, (2.11)

whereTw ⊆GF v, for any composite of valuations. Thus, ifT˜v(L|F)is trivial,

Tw˜ 'Tw◦˜ ˜v.

We will use three different types of fundamental groups, with the following notation:

1. π₁topis the topological fundamental group, the fundamental group of a fiber functor on the category of topological covers of a topological space.

2. πˆ1 is the profinite completion of π1top, the fundamental group of a fiber functor on the

category offinitetopological covers of a topological space.

3. π´et

1 is the étale fundamental group.

For a normal variety X over an algebraically closed subfield of _C, one has an equivalence be-tween the category of finite, étale covers of X and the finite, unramified covers of Xan, its corresponding analytic space over_C, by [GR57]. This leads immediately to the

Theorem 22(Comparison Theorem). LetX be a normal variety overCandx ∈ X(C).There

is a canonical isomorphism

ˆ

π1(Xan, x)'π1´et(X, x). (2.12)

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Let nowK =_CandF be a function field over_C. Then we have the following interpretation of Dv for a prime divisor. First, v is the valuation associated to a Weil prime divisor on some normal model X of F — that is, a normal variety with function field F, considered as a _C -scheme. Given X, there is a corresponding normal analytic spaceXan_{. Let}_X _{be a model of}_F

on which|v|is a prime divisor.

Example 23. The exceptional divisor E on Blp(X), the blowup at some closed point p of X,

gives a prime divisor on F but its center onX is not codimension 1and so is not centered as a

prime divisor onX.

LetD0 ⊆ X be the nonsingular locus of|v|;notice that the underlying topological space of

D0 is connected, as resolution of singularities gives an isomorphism with it and an open subset of a smooth algebraic curve.

Definition 24. LetN then be a normal disc bundle for (equivalently, a tubular neighborhood of)

D0 and

T =N \D0

the complement of D0 in its normal disc bundle N, which admits the normal bundle fiber se-quence

1 //F ι //T π //D0 //1, (2.13)

whereF is one of the fibers.

Letpbe a point onF. Note thatF, like all fibers of π, is a once-punctured disk and thus is homotopy equivalent to a circle. There is a surjection

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(proof: each normal étale cover of X gives a normal extension of its field of functions) whose kernel we will define to have fixed fieldFX, and the following commutative diagram:

1 //π1(ˆ F, p) ι //

&

ˆ

π1(T, p) π //

ˆ

π1(D0, π(p)) //1

π´et 1 (X, p).

(2.15)

Then we have

Proposition 25(The Geometric Theory of Decomposition and Inertia Groups). In the short exact sequence 2.15:

1. The top row is a central extension of groups, as the normal bundle is complex-oriented.

2. The image ofπˆ1(F, p)inπ1et´(X, p)is aTv(FX|F).

3. The image ofˆπ1(T, p)inπ1´et(X, p)is aDv(FX|F).

4. πˆ1(D0, π(p))is a quotient ofGF v, corresponding to covers ofD0 pulled back from covers

ofX.

We will denote bytv a generator ofπ1top(F, p) ⊆ π1(ˆ F, p),as well as any of its images in π´et

1 (X, p)orTv(FX|F).

Definition 26. We refer to such atv as ameridianofv.

Each meridian is almost unique — its inverse also gives a meridian ofv, albeit “in the opposite direction”. This should be viewed as a “loop normal to |v|”. Its image generates πˆ1(F, p). In

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Definition 27. LetΓbe a subgroup of a groupΠ. Then the abelianization functor gives a map

ab : Γab →π₁ab. (2.16)

We denote by Γa the image ofab. In particular, given v a valuation, andΠa quotient of GF or

π₁top(X)for some model of X, we will denote by T_va andDa_v the images of inertia and

decom-position, respectively, inΠab_{, which will sometimes appear in the sequel as}_H

1. We lettav be the

image of a meridian inΠab_.

We can also define the meridian of a valuationv on a modelX if|v|is smooth, and extend the definition to non-smooth |v| as follows. We resolve the singularities of |v| on X to get a birational map

η: ˜X →X

such that

1. ηis an isomorphism outside of|v|.

2. |v| ⊆X˜ is smooth.

Then we define a meridiantv onXto be

tv =η∗(tv) (2.17)

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following diagram: ˜ X00 ϕ ~ ~ ϕ0 ˜ X η ˜ X0 η0 ~ ~ X, (2.18)

we may always constructϕandϕ0 birational morphisms so that:

1. The above diagram commutes.

2. ϕ, ϕ0, η◦ϕ, η0◦ϕ0are isomorphisms outside of|v|.

3. |v|is smooth inX˜00.

In this case, the meridians inXdefined byη0 andηare the image of a meridian inX˜00underη◦ϕ

andη0◦ϕ0so, by commutativity of the diagram, the two meridians are the same.

We now describe the relation with composites of valuations. Letv be a prime divisor onF

andF v its residue field. Letwbe then a prime divisor on F v. A meridiantw inG(L˜v|F v)for some modelC ofF v then for each lift w˜ ofwtoL˜v thereto a uniquetw◦v inTw◦˜ v˜(L|F)and in

any model is a loop normal to|w|lying completely on|v|.

Definition 28. Ameridian of a rank-2Parshin chaintw◦v forL|F is defined to be such a lift.

As per Definition 27, any image in an abelianization will be denotedta w◦v.

LetF have dimensionn. Then ifv is ann−1-dimensional Parshin chain,F v is the function field of a curve overK. It is equipped with a fundamental, birational invariant: its unramified genusg(v). We can compute this as follows:

g(v) = rkˆ

ZDv

a_/_h_{T v}a_{, T}a

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2.2 Geometric Sets and the Maximal Smooth Model

Definition 29. We say that a setS of prime divisors of a function fieldF is ageometric set (of

F)if and only if there exists a normal modelXofF such thatS is precisely the set of valuations with centers Weil prime divisors onX. In this case, we write

S =D(X).

IfXis smooth, we sayX is amodelofS.

Theorem 30 (Pop). If F is a function field with K(F) = _Q, and let Γ ⊆ GF be a closed subgroup, up to conjugacy. Then there is a topological group-theoretic criterion, given one of the

representatives of Γ to determine whether there exists iand v ∈ Pari(F)such that Γ = Tv or

Γ =Dv.

This theorem is proven withGF replaced by the pro-`completion ofGF in [Pop11a]. To see this forGF as a whole, we apply Key Lemma 5.1 of [Pop11b]. Pop also proved [Pop11b]:

Theorem 31(Pop). Given a geometric setS of prime divisors onF,

1. If S is a geometric set of prime divisors onF, then a (possibly different) set S0

of prime

divisors onF is a geometric set if and only if it has finite symmetric difference withS.

2. There exists a group-theoretic recipe to recover

Geom(F) = {{(Tv, Dv)|v ∈ S} | Sa geometric set}.

Definition 32. IfS is any set of prime divisors onF, we define thefundamental group ofS to

be:

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Here, hTviv∈S is the smallest closed, normal subgroup of GF which contains every element in every conjugacy class in eachTv. IfT ⊆ S is a subset, then we denote by

ρT S : ΠT →ΠS

the restriction map, and drop subscripts when they are unambiguous.

Given a geometric set S, there are many possible X such that D(X) = S; for instance, any model less a finite subset of points has the same set of prime divisors. We now define the maximal model on which we will be able to effect our intersection theory.

Theorem 33. Let S be a geometric set for a function field F of dimension 2. There exists a unique modelM(S)ofF such that the following holds:

1. D(M(S)) =S.

2. M(S)is smooth.

3. πˆ1(Man, p)'π1´et(M(S), p)'ΠS.

4. IfXis any other smooth model ofF which satisfiesS =D(X), then there exists a unique

F-morphismX → M(S), and this is a smooth embedding.

Proof. LetU be a model ofS, and letX be a smooth compactification ofU. Let

∂ =D(X)\ D(U)

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1. LetX1 =X, ∂1 =∂.

2. We now construct (Xi+1, ∂i+1) from (Xi, ∂i). First, take the collection {vj} ⊆ ∂i such that each |vj| is a −1-curve such that no other |v0| that intersects it in the boundary is a −1-curve, and blow down. SetXi+1to be this blowdown, and∂i+1 =∂i\ {vj}.

As∂1is finite, at some point, this sequence becomes stationary — let’s say at(Xn, ∂n). Then we define

Umax =Xn\

[

v∈∂n

|v|. (2.19)

To prove that this satisfies property 4, letU0be another model andX0a smooth compactification of it. Run the algorithm onX0 to get a pair(X_n00, ∂_n00)Then by strong factorization for surfaces (see Corollary 1-8-4, [Mat02]), there exists a roof

Y

δ

~

υ

X_n00 X_n,

whereδandυare both sequences of blowups. For any morphism

ϕ:Z →Z0

of varieties we may define theexceptional locus

E(ϕ) =p∈Z0 |dim(ϕ−1(p))≥1

Letp∈ E(υ)∩Umax. Thenυ−1(p)is connected, soδ(υ−1(p))is also connected. It is proper, so

is either a union of divisors or a point. If it is a union of divisors and one of these divisors were in

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1theory. We may argue the same way forδ. Thus,δ◦υ−1|Umax\E(υ) is well-defined and regular

outside codimension2, so extends to a morphism

δ◦υ−1 :Umax→Umax0 ,

injective on closed points. By the same argument we may produce the inverse

υ◦δ−1 :U_max0 →Umax,

so we have that the maximal smooth model is indeed unique up to isomorphism. To prove property 3, we note that the map

U →Umax

gives a natural equivalence of the category of étale covers ofU with the category of étale covers ofUmax, by the Nagata-Zariski purity theorem [Gro05, X.3.4], and thus gives an isomorphism on

fundamental groups by Theorem 22.

Corollary 34. LetU be an affine or projective smooth variety with function fieldF. Then U =

M(D(U)).

Proof. If U is proper, the algorithm in the proof of Theorem 33 terminates immediately, so

Umax =U.

IfU is affine, letι : U → Umaxbe the embedding ofU as an affine open ofUmax. Assume

there were a a closed pointx∈Umax\U. Then there is an affine neighborhoodU0 ⊂Umaxsuch

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which contains x, and is not contained in U. Thus, Umax has a different codimension-1 theory fromU, which gives a contradiction.

Now, letF be a function field over_Q, and fix an embedding of_Qinto_C. Then ifXF is the inverse system of all smooth models ofF,

GF ' lim X∈XF

ˆ

π1((X×_QSpecC)an),

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Chapter 3 The Anabelian Intersection Theory

3.1 The Local Theory: The Intersection Theorem

The following theorem shows how the fundamental group detects intersections in the best-case scenario.

Theorem 35(The Local Anabelian Intersection Theorem). LetX be a smooth (not necessarily proper) surface over_C, and letC1 be a unibranch germ of an algebraic curve at a pointp∈ X

andC2an irreducible, reduced algebraic curve onX(a prime divisor), with distinct branchesγj

atp, with C1 distinct from each branch ofC2. LetY = X\C2 be the complement ofC2 inX;

this is an open subvariety ofX. Letw◦v1 be the rank-two Parshin chain corresponding toC1at p, and letv2 be the Weil prime divisor corresponding toC2. Then, we may choose meridianstav2

andta

w◦v1 inπ

´ et

1 (X\C2)ab

ta_w◦v₁ =i(p, C1·C2;X)tav2, (3.1)

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7.1]. Then

[Tv2(FX\C2|F)

a _:_T

w◦v1(FX\C2|F)

a_]_|_{i(p, C}

1·C2;X) (3.2)

with equality ifTv(FX\C2|F)

a_{is infinite in}_π´et

1 (X\C2)ab.

Lemma 36 (Reeve’s Lemma). We use the notation of Theorem 35. There is a neighborhood

B ⊆Xofpbiholomorphic to a unit ballB ⊆_C2_{about the origin so that}_p_{corresponds to}_(0,₀₎_.

Let∂Bbe the boundary ofB, which is homeomorphic to a3-sphere. Then for small enough such

B,

1. H1(∂B \(C2 ∩ ∂B),_Z) ' L

jZt a

γj, the direct sum generated by the meridian around

each branch, where each curve goes in the counterclockwise direction after identifying the

normal bundle with an open disc inC.

2. C1intersects∂Bin a loop`, which can be given an orientation by the complex orientation

onX. Then inH1(∂B\(C2∩∂B),Z), we have

` =X

j

i(p, C1·γj;X)taγj. (3.3)

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Proof. We prove the two assertions in order.

1. Embedded algebraic curves have isolated singularities, so eachγjintersects∂Bin a smoothly embeddedS1 _{(by the inverse function theorem), and each of these circles are disjoint. The}

Mayer-Vietoris sequence gives the first claim immediately.

2. Let

fj(x, y) = 0 (3.4)

be a local equation forγj, and let

u7→(x(u), y(u)) (3.5)

be a local parameterization of the normalizationC˜1ofC1atp. By [Ful98, Example 1.2.5b]

i(p, C1·γj;X) = vu(fj(x(u), y(u)), (3.6)

theu-adic valuation — or order of vanishing — offj(x(u), y(u))∈C[[u]].

Then the intersection with∂BofC1can be taken to be a pathη: [0,1]−→Cwith winding

number one around the origin so that

`(t) = (x(η(t)), y(η(t))). (3.7)

Let

πj :H1(∂B\(C2∩∂B),Z)−→Ztaγj (3.8)

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1-formdlogfj(x, y)restricted to∂B. Then

πj(`(t)) =

1 2πi

Z

`(t)

dlogfj(x, y),

onB, asdlogfj(x, y)is holomorphic away from the zero-set offj and otherwise measures the winding of a loop around the zero-set. But this pulls back to the contour integral

1 2πi

Z

η(t)

dlogf(x(u(t)), y(u(t)))

which evaluates exactly tovu(fj(x(u), y(u)))by the Residue Theorem.

Proof of Theorem 35. Let∂Bbe as in Lemma 36. Then given the map

ι:∂B\(C2∩∂B)−→X, (3.9)

we must compute

ι∗ :H1(∂B\(C2∩∂B),Z)−→H1(X,Z). (3.10)

But from Proposition 25, for eachj,

i∗(taγj) = t

a v2

and in the notation of Proposition 25,

`=ta_w◦v

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Thus by linearity, we have

ta_w◦v₁ =X

j

i(p, C1 ·γj;X)tav2 =i(p, C1·C2;X)t

a v2.

3.2 The Global Theory I: Points and Local Intersection

Num-bers

Definition 37. LetS be a geometric set.

1. Letv ∈ S. We define for every rank2Parshin chainp◦v the subset

∆(p◦v) =

  

 

w∈ S

∀Y ⊂ S finite, s.t.w∈Y andTw is torsion-free in

Πab

S\Y, Tp◦va 6={0}inΠabS\Y

     . (3.11)

2. We say p◦v ∼ p0◦v0 if and only if∆(p◦v) = ∆(p0 ◦v0), and the equivalence class of rank-two Parshin chains will be denoted by[p◦v]and called apoint. Given such a point

[p◦v], we will use∆([p◦v])to denote, for anyp0 ◦w∈[p◦v],∆(p0 ◦w).

3. We say that a prime divisorw∈ S intersectsa point[p◦v]ifw∈∆([p◦v]).

4. The set of points onS is denotedP(S).

5. Ifv ∈ SthenP(v)will be the set of points which contain an element of the formp◦v.

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Definition 38. LetY ⊂ Sbe a finite subset. Then we say thatY recognizesthe intersection ofv

atpifTa

v is torsion-free inΠabS\Y andY ∩∆(p) = {v}.

Definition 39. Let X be a smooth surface over an algebraically closed field of characteristic

zero. LetD1, D2 be two divisors onX. We define thetotal intersection product

(D1·D2) = X

p∈D1

i(p, D1·D2;X). (3.12)

Theorem 40(The Algebraic Inertia Theorem). In this theorem (and its proof), all homology will

be taken with integral coefficients. Let {Di}i∈I be a finite collection of smooth, distinct prime divisors on a smooth, proper, complex, algebraic surfaceX such that

D=[

i∈I

Di (3.13)

is simple normal crossing. Let

η1 : NS(X)→

M

i∈I

Ztavi (3.14)

be given by

η1(D) =

M

i∈I

(D· |vi|)tavi. (3.15)

Then there is a short-exact sequence

NS(X) ∂ //L

i∈IZt a

vi //ht

a

viii∈I ⊆H1(X\D) //0.

(3.16)

Proof. For eachi∈I let

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be an open normal open square bundle ofDiinXwith projection to its central divisor. Then

Ti =Ni \Di (3.18)

is an open subset ofX, as isNi. For eachi∈Ilet

Pi =

[

j∈I,j6=i

Di∩Dj

!

⊂Di andP =

[

i∈I

Pi. (3.19)

We further shrink the square bundles so that each component of each intersection betweenNi’s contains exactly one pointp∈P. Call each componentNp, identified by the point of intersection it contains. We can then further shrink the square bundles so that

p6=q =⇒ Np∩ Nq =∅. (3.20)

Let also

Di,p =πi(Np). (3.21)

By the assumption that D is simple normal crossings, andp ∈ Di ∩Dj where i 6= j, we can assume that there is an analytic isomorphism

ηp :Np ' {(x, y)∈C2 | <x,<y,=x,=y ∈(−1,1)}, (3.22)

so that

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and so that the diagrams

Np ηp // πi|Np

{(x, y)∈_C2 _{| <}_x,_<_y,₌_x,₌_y_∈₍₋_1,₁₎_}

π1

Di,p

ηp _/_/

{x∈_C| <x,=x∈(−1,1)},

(3.24)

and

Np ηp _/_/

πj|Np

{(x, y)∈_C2 _{| <}_x,_<_y,₌_x,₌_y_∈₍₋_1,₁₎_}

π2

Dj,p

ηp _/_/

{x∈_C| <x,=x∈(−1,1)}

(3.25)

commute, whereπ1 andπ2 are projection onto the first and second coordinates, respectively. We

then set

D0_i =Di\Pi. (3.26)

and letT_i0 be defined as a fiber bundle overD_i0 with structure mapπ_i0 by the pullback square

T0 i //

π0_i

Ti πi|T_i

D0_i //Di.

(3.27)

We now define for eachp∈P withp∈Di∩Dj andi6=j

Tp =Np\(Di,p∪Dj,p). (3.28)

We now define for eachi∈I, p∈Pi:

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can intersect atp.

2. T+

p = π 0−1

i (D

+

i,p)∪π 0−1

j (D

+

j,p). Note thatTp ⊂ Tp+ is an open subset. Again, we modify ourD_i,p+ so that the distinctT+

p are mutually disjoint.

3. Ui =Ti\S

p∈Pi(πi |Ti)

−1 (Di,p)

.

4. D−_i,p =D+_i,p\Di,p andTi,p−=π 0−1

i (D − i,p).

Then formula 3.22 gives us a pushout-pullback square of open subsets

`

i∈I,p∈PiT

− i,p

ι1 //

ι2

`

p∈PTp+ ι3

`

i∈IUi

ι4 //

T.

(3.29)

We choose a fiberFi of eachTi0and let

i :Fi −→ Ti (3.30)

be its inclusion, and note that

H1(Fi) = Ztavi. (3.31)

We now set

T =[

i∈I

TiandN =[

i∈I

Ni, (3.32)

and we have a pushout-pullback square of open subsets

T ι1 //

ι2 N ι3

X\D ι4 //X

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Let

η1 :H2(X)−→

M

i∈I

H1(Fi)andη2 :

M

i∈I

H1(Fi)−→H1(T) (3.34)

be given by

η1(γ) =

M

i∈I

(γ∩Di)tvi andη2 =

M

i∈I

i∗. (3.35)

We have a diagram:

L

i∈IH1(Fi) η2

H2(X)

η1

8

∂ _/_/

H1(T)

ι1∗+ι2∗ //

H1(N)⊕H1(X\D)

ι3∗−ι4∗//

H1(X).

(3.36)

The bottom row is exact, by the Mayer-Vietoris sequence applied to diagram 3.33. We prove the following, the combination of which implies the exactness of the short exact sequence 3.16:

1. We can apply the Mayer-Vietoris sequence to the diagram 3.29, and we get

L

i∈I,p∈PiH1(T

− i,p)

ι1∗+ι2∗//L

p∈P H1(Tp)

⊕ L

i∈IH1(Ui)

ι3∗−ι4∗//

H1(T)

∂T _/_/

L

i∈I,p∈PiH0(T

− i,p)

(3.37)

This allows us to computeH1(T)as follows.

(a) The K¨unneth formula gives us that

H1(Ti,p−) = Zt a

vi⊕H1(D

−

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and ifp∈Pj then we may use Lemma 36 to give us

H1(D−i,p) =Zt a

vj. (3.39)

(b) The bundleT0 i toD

0

i is complex-oriented so has trivial monodromy action. But

H2(D0_i) = 0, (3.40)

so every such bundle must be trivial.Uiis thus a trivial bundle overDi\

S

p∈PiDi,p

which is a deformation retract ofD0_i. Thus, by the K¨unneth formula again,

H1(Ui) =Ztavi⊕H1(D

0

i) (3.41)

(c) Let p ∈ Pi ∩Pj where i 6= j. Then Tp is a product, by diagram 3.22, and by the K¨unneth formula again:

H1(Tp) = Ztavi ⊕Zt

a

vj. (3.42)

We may run a similar — but simpler — analysis forN. Plugging this information into the Mayer-Vietoris sequence 3.37, we get a commutative diagram

0 // L

i∈IH1(Di)

⊕ L

i∈IZtvi

/R

Φ

αT _/_/

H1(T)

∂T _/_/ ι1∗

L

i∈I,p∈PiH0(T

− i,p)

0 // L

i∈IH1(Di)

αN //

H1(N)

∂N _/_/L

i∈I,p∈PiH0(T

− i,p)

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of relations

R =

* X

p∈Pi

p∈Di∩Dj tvj

+

i∈I

. (3.44)

Chasing the diagram, we see that

imαT

M

i∈I

Ztvi

!

/R

!

= imη2. (3.45)

2. We now wish to show that there existsη1 such that

∂ =η2◦η1, (3.46)

and thatη1is defined as in formula 3.15. We note that if indeedη1satisfies these properties,

thenR⊆imη1, so these are all the relations given in 3.16. But by Diagram 3.43,

kerι1∗∩kerι2∗ ⊆kerι1∗ ⊆imη2. (3.47)

Thus,∂ factors through the image ofη2, and we call this mapη1.

To compute η1, we first use the orientation class [ω] ∈ H4(X) to give the intersection

∩-product pairing

∩:H2(X)OH2(X)→_Z. (3.48)

Let c ∈ H2(X)have realization a connected, closed topological surface |c|, which inter-sects eachDi transversely in finitely many points{Qj}j∈J with orientations

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away from Pi. Such elements generate H2, so we only need to verify the identity for them. We recall that for a space Y, Ci(Y)is theZ-module of singular chains ofY with

coefficients in_Z. We then have, from the proof of the Mayer-Vietoris sequence, a diagram

C2(N)⊕C2(X\D)

ι3∗−ι4∗// ∂1⊕∂2

C2(X) //

∂3

0

C1(T)

ι1∗+ι2∗//

C1(N)⊕C1(X\D)

ι3∗−ι4∗//

C1(X) //0

(3.50)

LetN0

i ⊂ Nibe a slightly smaller tubular neighborhood ofDi, and let

N0 ₌[

i∈I N0

i . (3.51)

We recall the definition of the boundary map∂ in the Mayer-Vietoris long exact sequence: we take˜cto be a preimage of the cycle representingcbyι3∗−ι4∗,which is surjective. Then

∂1(˜c)lies in the kernel ofι3∗−ι4∗by commutativity of the diagram, so lies in the image of

ι1∗+ι2∗, because∂3(c) = 0. We can then choose the lift so the assertion is clear. We can

take

˜

c= (|c| ∩ N0₎_{⊕ −}₍_|_c_{| ∩}_(X_{\ N}0_)),

(3.52)

where these intersections stand for their representations as2-cycles. But∂1(|c| ∩ N0)is a

boundary, as|c| ∩ N0 _{is a union of closed disks. Thus, the only contribution in homology}

comes from∂2. But we can compute

∂(|c| ∩(X\ N0)) =− X j∈J,Qj∈Di

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This is exactly

X

i∈I

−(c∩Di)tvi. (3.54)

However, as each Di is algebraic, the Hodge-Lefschetz (1,1)-theorem says that this map factors throughNS(X), which finally proves the exactness of the sequence 3.16.

Corollary 41 (The Separation of Inertia Criterion). We preserve the notation of Theorem 40. Suppose that, in addition to the hypotheses, for each i, j ∈ I there does not existE ∈ NS(X)

s.t. for everyk ∈I\ {i, j},

(E·Dk) = 0, but(E·Di)6= 0or(E·Dj)6= 0. (3.55)

For eachi∈I, letvibe the prime divisor onC(X)associated toDi. Then inH1(X\D),

1. Ta vi 'Z.

2. Forj ∈I, j 6=i,

T_va_i∩T_va_j = 0. (3.56)

Proof. We apply the hypothesis in 3.55 to 3.16 to see that there are no elements in the image of

∂ none which are zero at all but one component to prove the first claim, and which are zero at all but two components to prove the second claim.

Lemma 42 (Divisor Existence Lemma). LetD1, . . . , Dj be a finite set of divisors on a smooth surface, and p1, . . . , pnbe a finite set of points. Then there are infinitely (in fact “generically”)

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Proof. Choose a very ample divisorC. ThenC·Di >0for alli. Then having an intersection at

pi is a closed condition (sinceC is basepoint-free), and so the divisors which do not intersect at those points form an open, nonempty subset of the linear system|C|, which is then infinite.

Corollary 43. LetS be a geometric set and letP ∈ M(S). Letv ∈ S and letp◦v be a rank-2

Parshin chain such that

|p◦v|=P. (3.57)

Then

1. w∈∆(p◦v)if and only ifP ∈ |w|.

2. Ifw ∈ ∆(p◦v)then there exists a setY ⊂ S which recognizes the intersection ofwand

[p◦v].

3. Ifv0 ∈ S andp0◦v0 is not centered onM(S), then∆(p0◦v0)6= ∆(p◦v).

Then there is a canonical map

ι:M(S)−→ P(S), (3.58)

given by

ι(x) = {v ∈ S |x∈ |v|}. (3.59)

This map is always an injection, and is a bijection if and only ifM(S)is proper.

Proof. LetX be a smooth compactification ofM(S)with only simple normal crossings at the boundary and let

∂S = Div(X)\ M(S)⊂Par1(F) (3.60)

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1. Let P ∈ |w| and Υ ⊂ S a cofinite subset which does not contain w and so that T_wa is torsion-free inΠab

Υ. Then Lemma 36 implies that the projection ofTp◦va to theTwa-part of the direct summand is torsion-free in Ta

w in any ΠabΥ, so must be torsion-free itself, so w

satisfies the properties of an element of∆(p◦v).

2. Lemma 42 shows that there exists a finite set{βi}i∈Iof prime divisors such that{|βi|}i∈I∪ {|w|}∪{|β|}β∈∂Ssatisfies the hypotheses of Corollary 41, and so{βi}i∈I∪{w}recognizes

the intersection ofwwithp◦v.

3. This follows immediately from the fact that there exists a prime divisor in M(S) which intersects the boundary at|p0◦v0|and does not intersect|p◦v|, so the valuation associated to this divisor will be in∆(p0◦v0)but not in∆(p◦v).

The existence and injectivity of the map ι is now straightforward. The bijectivity in case of properness is the valuative criterion for properness.

Definition 44. LetS be a geometric set,v, w ∈ S, andp◦wa rank-2Parshin chain. Then we define thelocal intersection number(p, v·w;S) as follows:

1. If there does not exist a setY which recognizes the intersection ofv atp(and this includes

the case wherev /∈∆(p), then we define

(p, v·w;S) = 0. (3.61)

2. Otherwise, letY recognize the intersection ofvatp. Then we define

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Theorem 45. LetS be a geometric set,p∈ P(S)a point with a center|p| ∈ M(S)andv ∈ S. Then

X

p0_{◦w∈p,w∈S}

(p0, v·w;S) =i(|p|,|v| · |w|;M(S)). (3.62)

Proof. By Corollary 43, either both sides of Equation 3.62 are zero, or there exists a geometric set which recognizes the intersection ofpwithv. Now, eachp0◦w∈prepresents a branch of|w| atp, and we will call this germξp0_◦w. We then have by the Local Anabelian Intersection Formula

i(|p|,|v| · |w|;M(S) = X

|p0_◦w|∈p

i(|p|,|v| ·ξp0_◦w;M(S)) =

X

|p0_◦w|∈p

(p0, v·w;S). (3.63)

Definition 46. Letp ◦v be a rank-2 Parshin chain with v ∈ S. Then we say that p◦v is a

nonnodal chain for S if [p◦v]is distinct from every other [p0 ◦v], and we say that p◦v is a

noncuspidal chainif there exists av0 ∈ S with(p, v0 ·v;S) = 1. A rank-2Parshin chain which

is both nonnodal and noncuspidal will be called asmooth chain.

Proposition 47. If|p◦v| ∈ M(S)thenp◦vis nonnodal (resp. noncuspidal) if and only if|v|is not nodal (resp. cuspidal) at|p◦v|.

Proof. This follows directly from Corollary 43 and the definition of a nodal, resp. cuspidal, point on a curve.

Recall thatP(S)⊂2Par2(F)_{; thus, each point of each geometric set is a subset of a larger set,}

and we will consider them as sets in the next sequence of definitions.

Definition 48. LetS0_{be a geometric set, and let}_p_{∈ P}₍_S0₎_{. Then we define the}_S0_{-limits of}_p_in S by

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Definition 49. Ifv ∈ S0_{, then we define the}_S0_{-limits of}_v_in_S_{to be}

LimS(v) =

[

p◦v∈Par2(v)

LimS([p◦v]). (3.65)

Definition 50. LetS0

be a geometric set so thatS ⊂ S0

. Then we define theboundary points of

S relative toS0 as

∂S0(S) =

[

v∈S0_\S

LimS(v), (3.66)

and theinterior points ofS relative toS0 as

PS0(S) =P(S)\∂_S0(S). (3.67)

Definition 51. We say that a point[p◦v]isabsolutely uncenteredonS ifTp0_◦v0 is nontrivial in the total fundamental groupΠS ofS for somep0◦v0 ∈[p◦v]. We define

a(S) = {[p◦w]∈ P(S)|[p◦w]absolutely uncentered}.

We define thecandidate points ofS by

A(S) =P(S)\a(S),

and these are the points which are not absolutely uncentered.

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Proposition 52. For any geometric setS,

M(S)⊆ A(S).

That is, absolutely uncentered points of S do not have centers onM(S), and candidate points have a chance.

The converse to this proposition is false in general:

Example 53. LetF =_Q(x, y), andX = Spec_Q[x, y]andS = Div(X). Ifp◦vis not centered onX, then the algebraic inertia theorem (accounting for resolution of singularities) shows that

Ta

p◦v =Zin any divisor complement, so

∆(p◦v) = S. (3.68)

But asΠS is trivial,

A(S) =P(S). (3.69)

Thus,

A(S) =X∪ {∞}. (3.70)

However, for the “visible affine opens” we define below, the converse is true. The first goal of Anabelian Intersection Theory is to identify these special geometric sets, and use this to construct a salvage of the converse.

Anabelian Intersection Theory