U NIVERSITÀ DEGLI S TUDI DI U DINED IPARTIMENTO DI M ATEMATICA E I NFORMATICA
D OTTORATO DI R ICERCA IN M ATEMATICAP H .D. T HESIS

Geometric Lang-Vojta
Conjecture in P2

C ANDIDATE :S UPERVISOR :Amos Turchet

Prof. Pietro Corvaja

Academic Year 2013

Typeset in LATEX.
Self-published in May 2014.
Copyright c 2014, Amos Turchet.

Author: Amos Turchet
Address: Dipartimento di Matematica e Informatica
Università degli Studi di Udine
Via delle Scienze, 206
33100 Udine
ItaliaE-mail: amos.turchet@uniud.it
Web Page: http://sole.dimi.uniud.it/~amos.turchet/Knowing,
my most esteemed friend Dionysius,that you are anxious to learn
how to investigate problems in numbers,
I have tried [ . . . ] to set forth to you
the nature and power
subsisting in numbers.
Diophantus - Arithmetica

Abstract
Lang-Vojta conjecture is one of the most celebrated conjectures in Diophantine Geometry. Stated independently by Paul
Vojta in [Voj1] and Serge Lang (see [Lan3]), the conjecture predicts degeneracy of S-integral points in algebraic varieties oflog-general type for a finite set of places S of a number field κcontaining the infinite ones, provided that the divisor “at infinity” is a normal crossing divisor. This deep conjecture andhis analogous formulations are among the main open problems in Number Theory, Complex Analysis and Arithmetic
Algebraic Geometry.
This thesis contains the work of the author during his Ph.D.
studies at the University of Udine under the supervision ofProf. Pietro Corvaja (and, partially, during his visit to Brown
University under the supervision of Prof. Dan Abramovich),
and it is centered around the function field version of LangVojta conjecture for complements of curves in P2 , with at mostnormal crossing singularities. The main part contains theproof of two cases of this conjecture, namely the non-split
case for complements of degree four and three components
divisors and the split case for very generic divisors of degree
four with simple normal crossing.
After a detailed review of the main conjecture, to which isdevoted the first part, in the second part we deal with an extension of previous results of Corvaja and Zannier [CZ5] forcomplements of a conic and two lines to the so-called non-

split case. We focus on the situation of an affine threefold
fibered over a curve and we study sections of fibrations in which every fiber is isomorphic to the complement of a conic andtwo lines in P2 . We describe completely the moduli of degree
four and three components divisors in the projective plane;
this permits us to link the problem to the study of solutions toa certain equation with non constant coefficients in the function field of the base curve. Then we implement methods relying on S-unit gcd for function fields and height considerations already available for the split case and we adapt themto our case obtaining algebraic hyperbolicity for sections ofthese fibrations.
The third part is dedicated to the case in which the divisor at
infinity has less than three irreducible components. We adopt
a new strategy to tackle this problem based on ideas coming from logarithmic geometry. We begin extending previous
result for the three component case in the language of logarithmic stable maps. Using this extension we reformulate theproblem as vanishing of certain moduli space of minimal stable maps KΓ (P2 , D ) in the sense of [AC]. Finally we use theproperness of the stack KΓ (P2 , D ) of minimal logarithmic stable maps to the Deligne-Faltings pair (P2 , D ) in order to obtain algebraic hyperbolicity for complements of very genericquartics D deforming to a conic and two lines in P2 . This provides first examples of Lang-Vojta conjecture for complementof both irreducible quartics and two conics, although a very
generic hypothesis is needed.

Acknowledgments
Traveler, the road is only
your footprint, and no more;traveler, there’s no road,the road is your traveling.
Going becomes the roadand if you look backyou will see a pathnone can tread again.
Antonio Machado - Proverbios y cantares (XXIX)
The journey to the end of this Thesis has been long and challenging but I had the luck of sharing it with so many supporting and inspiring people. It is a pleasure to thank (someof) them who helped me, knowingly and unknowingly, in the
successful completion of this project.
First of all I would like to express my special thanks to my supervisor Pietro Corvaja who has introduced me to the beautyof Numbers and showed me the wonderful landscapes ofArithmetic; he has always provided insightful discussions about the research and has given me the freedom to pursue various projects providing essential help in these three years. I
am also very grateful to all the other geometers in Udine, inparticular Francesco Zucconi for many long and fruitful discussions and for always broaden my views, Pietro de Poi forformulas, coffees, and solutions of doubts and Paolo Cragnolini who introduced me to Geometry with the capital "G". I
would also like to thank all the professors and administrative
personnel of DIMI for these ten years from undergraduate todoctoralship.
A special thank goes to Barbara Fantechi and Carlo Gasbarri
who kindly accepted to be in my Ph.D. committee and withwhom I had very fruitful conversations and discussions.
I had the fortune to share my office with friends more thancolleagues: to Daniele, Edda, Emanuele and Luca go my gratitude for laughs, shared (and divided) food, happy moments,
coffees and the (small amount of) math we made together inthese years. At the same time my university life has beenenriched with the presence of many good friends: Sara has
been and still is the best friend I have ever met and Tommaso
has always brought that lighthearted joy to my work days (although he tried to stop my Thesis with Minecraft c !).
In these three years I have been so lucky to have the opportunity to spent almost 6 months at Brown University, in Providence (RI - USA). I am in debt with Dan Abramovich for making my stay in Providence possible, and for adopting me as“half-student” during those months: his support and his precious hints have played a big role in my mathematical development. At the same time I want to sincerely thank Joe Silverman for accepting to supervise me in that period and for thevery interesting discussions. I would also like to thank all thepeople who helped me with the bureaucracy and with all myneeds at the Math Department of Thayer Street, in particular
Audrey, Doreen and Lori.
In the same days I met many people who turned my stay in
the US into a sunny, joyful period. Above all my office-mate,music-mate, AG-mate, DMB-mate and food-mate Kenny for
uncountable discussions on math, music and food: you havebeen a real friend and I owe you so much for all the happinessof those months. I would like to thank Amalia, Hoang, Ian,

Martin, Nhat, Sam, Asaf and all the students in the department who shared with me good moments in the basement
and outside. I also thank all the members of the Brown Chorus for all the good music and vibes and for a wonderful Messiah together.
I also met a lot of people during this world-wide trip, whoshared their math, their passions and their food with me: I
am almost sure I am forgetting many of you, but I really thankyou for the gifts you have left to me: Pranabesh, Marco,
Francesco, Roger, Liana, Charles, . . . , and in particular Natalie.
I want to thank, from the deep of my hearth, my whole Family: my mum Patrizia, who raised me almost by herself, providing unconditional love and care in any single moment of
my life, and my brother Damiano, who shared with me all thegood and bad moments, rejoicing for the former and supporting me in the latter. I love them both so much, and I would
not have made it this far without them. To my enlarged family, my “as-brothers” Massimo and Federico, my aunt Pierina,
Alice, Antonella, Mia, Silvia and the little Nicolò, thank you
for being as you are. I want to thank also all my uncles, myaunts and my cousins too.
Last but not least, I would like to thank all my music brothers
and sisters: to those who have played with me and those who
still share their music with me, to the crazy Confusion crew,to the unknown listener and the famous player, to the songs
and the lyrics who moved me in any way, thank you because“without music, life would be a mistake”.

Contents
Introduction
I.1 What is in this Thesis . . . . . . . . . . . . . . .
I.2 What is not in this Thesis . . . . . . . . . . . . .

v
ix
xii

I

Lang-Vojta conjecture11Lang-Vojta for Number Fields
1.1 Integral points on Curves . . . . . . . . . . . .
1.1.1 Siegel’s Theorem . . . . . . . . . . . . .
1.1.2 Mordell Conjecture . . . . . . . . . . . .
1.2 Bombieri-Lang and Lang-Vojta Conjectures . .
1.2.1 From Curves to Surfaces . . . . . . . . .
1.2.2 Bombieri-Lang Conjecture . . . . . . . .
1.2.3 Extensions and Lang-Vojta Conjecture .
1.3 New techniques from Schmidt’s Subspace . . .
1.3.1 Schmidt’s Subspace Theorem . . . . . .
1.3.2 Lang-Vojta for many components . . .
1.3.3 A new proof of Siegel’s Theorem . . . .

3
4
5
8
11
13
16
19
23
24
24
26

2

Function Fields
2.1 Geometric Mordell . . . . . . . . . . . . . . . .
2.1.1 Function fields and Number fields . . .
2.1.2 Mordell Conjecture for function fields .
2.2 Geometric Lang-Vojta . . . . . . . . . . . . . . .
2.2.1 Integral models and Lang-Vojta Conjecture . . . . . . . . . . . . . . . . . . . . .
2.2.2 Algebraic Hyperbolicity . . . . . . . . .

29
30
30
33
36
36
39

ii

Contents

II

Three components case43

3Known results
3.1 Large number of components . . . . . . . . . .
3.1.1 The four line case . . . . . . . . . . . . .
3.2 The split case . . . . . . . . . . . . . . . . . . .
3.2.1 The conic and the two lines . . . . . . .
3.2.2 Greatest Common Divisor estimates . .
3.3 Extension to ramified covers of G2m . . . . . . .
3.3.1 Extra divisor from ramification . . . . .

45
45
47
48
50
53
55
57

4

The non-split case
4.1 Statement of the problem . . . . . . . . . . . . .
4.2 Moduli of three components divisors . . . . . .
4.3 Affine threefolds . . . . . . . . . . . . . . . . .
4.4 Proof of the result . . . . . . . . . . . . . . . . .

59
59
62
65
66

III

The split case85

5Approaches to fewer components
5.1 Limits of Corvaja and Zannier methods . . . .
5.2 Ideas from deformations and curve counting .
5.2.1 Chen’s argument . . . . . . . . . . . . .
5.2.2 Limits of Chen methods . . . . . . . . .

87
88
90
91
95

6

Logarithmic Geometry99
6.1 Background and basic constructions . . . . . . 100
6.1.1 Log structure and log-schemes . . . . . 101
6.1.2 Log maps . . . . . . . . . . . . . . . . . 105
6.1.3 Log smoothness . . . . . . . . . . . . . . 108
6.2 Log stable curves . . . . . . . . . . . . . . . . . 110
6.2.1 Log curves as stable curves . . . . . . . 110
6.2.2 Canonical log structure on stable curves 113
6.3 Log stable maps . . . . . . . . . . . . . . . . . . 116
6.3.1 Basic properties of log stable maps . . . 117
6.3.2 Deligne Falting pairs and moduli of minimal log stable maps . . . . . . . . . . . 121

Contents

7iii

Complements of a very generic quartic
127
7.1 Three components case for log-stable maps . . 128
7.1.1 Extending Corvaja and Zannier Theorem 128
7.1.2 Extension to log stable maps . . . . . . 133
7.2 Emptiness of moduli spaces . . . . . . . . . . . 139
7.2.1 Algebraic Hyperbolicity via moduli of
log-stable maps . . . . . . . . . . . . . . 141
7.3 Properness of the stack . . . . . . . . . . . . . . 144
7.3.1 Invariance under deformation . . . . . 146
7.3.2 Properness of the moduli space of logstable maps . . . . . . . . . . . . . . . . 148
Bibliography

151

iv

Contents

Introduction
Diophantine Geometry can be described as the study of integral or rational solutions to polynomial equations with integral or rational coefficients, the so-called Diophantine Equations, using geometric tools. In its simpler instance it asks fora “geometric” solution to the following
Problem (see [Mor2]) Find reasonably simple necessary conditions for the solvability off ( x1 , . . . , x n ) = 0
for a polynomial f with coefficients in Q, with ( x1 , . . . , xn ) ineither Zn or Qn .
The term “geometric” stands for depending on the geometricproperties of the complex algebraic variety defined by f , i.e.
the set of complex solutions to the previous equation. Moreprecisely and more generally, fixed a field κ (which usually iseither a number field, or a function field of an algebraic variety), we can describe Diophantine Geometry as the studyof the set X (κ ) for an algebraic variety X over κ. Rephrasing
Serge Lang, who coined the term for this discipline, one cansay that the main goal of Diophantine Geometry is to determine geometric properties of the quasi-projective variety X
that characterize the set of points X (κ ), e.g. that imply that
X (κ ) is either non-empty or finite or dense with respect to the
Zariski topology. Here geometric means properties that can bechecked on the algebraic closure of κ.
While the study of Diophantine Equations is one of the oldest
branches of mathematics which can be traced back to Babylonians and Egyptians, Diophantine Geometry is a fairly recent

viIntroduction

subject whose name comes from the seminal first edition ofLang’s book [Lan3]. From its birth numerous results on the
behavior of sets X (κ ) have been achieved, but at the sametime, it seems that a large amount of open problems and conjectures are still out of reach.
One of the most intriguing and influential open problems inthis field is to determine whether there exist infinitely many
integral points on algebraic varieties defined over Q, or equivalently infinitely many integral solutions to the system of
equations defining the variety. Since the Siegel Theorem, andlater Falting’s proof of Mordell Conjecture, this task has been
completed for the case of algebraic curves. In particular, it has
been shown that whenever an affine curve has genus greaterthan zero it contains only finitely many integral points. For algebraic surfaces the problem turns out to be much more subtle and challenging and, although deep results have recently
been obtained, a complete solution seems to be, at present,
beyond hope. Nevertheless, a number of conjectures havebeen stated and serve as both focal points as well as the direction towards which Diophantine Geometry research is moving. Among these conjectures, one of the most important, andthe one to which this Thesis is devoted, is the conjecture due
to Paul Vojta [Voj1] and Serge Lang [Lan3] which for number
fields (in the surface case) reads as follows:
Conjecture 1 (Lang-Vojta for number fields). Let X be asmooth affine surface defined over a number field k. Let X̃ be asmooth projective variety containing X as an open subset. Let D =X̃ \ X be the divisor at infinity and K = KX̃ be a canonical divisorof X̃. Suppose that D is a normal crossing divisor. Then if D + K isbig, for every ring of S-integers OS ⊂ k, the set of S-integral points
X (OS ) is not Zariski-dense.
The history of this Conjecture can be traced back to the workof Vojta in [Voj1], where he formulated several conjectures
on distribution of integral, S-integral, and algebraic points on
projective varieties which «seem to contain virtually all Dio-Introductionvii

phantine statements that are currently proven or conjectured»[HS]. Independently Lang proposed a series of conjectures on
the same themes conjecturing a unified description of the socalled exceptional set on log-general varieties over number andfunction fields and its analogous in Complex Analysis (see[Lan1], [Lan2], [Lan3] and [Lan4]). These two profound visions of Number Theory (and Complex Analysis) meet together in the aforementioned Conjecture which we called, as
customary, with both names.
Conjecture 1 extends Bombieri-Lang Conjecture on rational
points on general type varieties and can be regarded as oneof the biggest open problems in Arithmetic Geometry. Unfortunately only some special cases of the Conjecture are known,
and a complete solution at the present seems to be out ofreach. In particular, almost all the results follows from Falting’s Big Theorem, which proves a famous conjecture by Langon the non-density of rational points for irreducible subvarieties of abelian varieties which are not a translate of an abeliansubvariety.
A slightly easier problem to tackle is the one obtained by theprevious Conjecture by replacing the number field k with afunction field in one variable κ (C), i.e. the function field ofan algebraic curve C , and the set S by a finite set of points of
C . In this setting, one can try to use geometric tools not available in the number field case, like the existence of derivatives
and differential forms canonically defined on the curve C , toobtain stronger and broader results. Moreover, the similarity of this approach with problems related to the distributionof entire curves in complex analytic manifolds allows one toborrow ideas and techniques from this related field.
In the last ten years, following a new proof of Siegel’s Theorem using Schmidt’s Subspace Theorem, Corvaja and Zannier were able to prove several special cases of Lang-Vojta’s
Conjecture in the case in which the divisor at infinity has sufficiently many irreducible components. Among these results,

viii

Introduction

when the compactification of the affine surface X is the projective plane P2 , they solve the Conjecture (the so-called splitcase) provided that the divisor D has at least three components. This seems to be the most general situation in which their method can be applied for complements of normal
crossing divisors in P2 . In particular this leaves open thecase of log-irregularity strictly lesser than two. Using a different strategy, Xi Chen proved that the complement of a very
generic plane curve, with at most normal crossing singularities, of degree at least 5, verifies Conjecture 1. His result, moving from different motivations, related to the famous
Kobayashi Conjecture, is based on a very refined deformationargument which reduces the general situation to the known
case for complements of hyperplanes in general position. Similarly as before, it seems that his techniques cannot be brought
further to the complete proof of the Conjecture in the caseX̃ = P2 .
The present Thesis focuses on two open problems related tothese results. The first one is a generalization of Corvaja andZannier’s result for complements of a conic and two lines ingeneral position in P2 . In particular we considered Conjecture 1 in the non-split case, i.e. when the conic and the lines
are not defined on the base field. This led to consider fibered
threefolds in which every fiber is isomorphic to the complement of a degree four divisor with three irreducible components. In this situation we were able to prove Lang-Vojta Conjecture, and explicitly determine a bound for the degree ofaffine curves in terms of their Euler Characteristic. Moreover
we calculate the dependency of this bound on the structureof the threefold, based on a description of the moduli spaceof the conic and two lines in P2 .
The second problem we deal with in this Thesis is the complete solution of Conjecture 1 (in the split case) for complements of plane curves in P2 . Our approach unifies the aforementioned methods, Corvaja-Zannier and Chen, obtaining
Lang-Vojta Conjecture for complements of very generic quar-

What is in this Thesisix

tics in the projective plane, which in particular completes theproof for affine varieties of log general type whose compactification is P2 , although with a very generic hypothesis which is present neither in the original Conjecture, nor in Corvaja and Zannier’s results. We used a deformation argument
which reduces the general case to an extension of Corvaja
and Zannier’s result in the three components case. However,
although the deformation argument is similar to the workof Chen, our strategy is different, making use of Logarithmic Geometry, in particular the Theory of logarithmic stablemaps to Deligne-Faltings pairs, as developed by Chen and
Abramovich-Chen. It turns out that this approach led to asimpler solution of the problem despite the fact that the reformulation for log-maps is more technical and requires refinedtools.

I.1

What is in this Thesis

Since the problem we deal with lies in the intersection of
Number Theory and Algebraic Geometry, we tried to makethe exposition as self-contained as possible, assuming only
basic knowledge of both fields, at the level of [Hin2] and[Har]. We hope in this way that both number theorists andalgebraic geometers, can follow the parts that are more arithmetic in nature, as well as the ones that have more geometricflavor. We sketch here, for the reader’s convenience, the planof the Thesis.
The presentation is divided in three parts which can, a priori, be read separately. The first part is devoted to a throughout discussion of the origin and features of Lang-Vojta Conjecture. It contains, in particular, most of the background inArithmetic Geometry, needed to understand the subsequentparts. At the same time it describes the motivations and thehistory of the Conjecture together with all the results and theideas related to this fascinating subject. The second part is fo-

xIntroduction

cused on the non-split case of Conjecture 1 for complementsof degree four and three components divisors. Finally theThird part concerns the general case of Geometric Lang-Vojta
Conjecture for complements of very generic plane curves ofdegree four with at most normal crossing singularities. Together with the main result it contains also an introduction to
Logarithmic Geometry and to the results that we use in our
proof.
We present here a more detailed description of the contents ofeach chapter.
The first chapter deals with the number field case, movingfrom the one dimensional classification of algebraic varieties
from a Diophantine point of view, and discussing the problem
for surfaces with an analysis of Bombieri-Lang Conjecture.
After this first part the chapter treats the extension due to
Vojta and his conjectures, with some height machinery background, together with the relationship between previous Conjecture and Vojta ones. The last section is devoted to a description of the new method developed by Corvaja and Zannier
from their new proof of Siegel’s Theorem and its applications
to the study of integral points on surfaces.
The second chapter is devoted to the function field case ofmain Conjecture. Similarly to the first chapter it contains a
discussion of the one dimensional situation related to the famous Geometric Mordell Conjecture, which serves as an introduction to function fields arithmetic and to the peculiarfeature of isotriviality. Then the chapter contains a reformulation of Lang-Vojta Conjecture using the theory of integral
models of algebraic varieties, which gives a unified vision of
the two sides of the Conjecture. The last part of the chapter isdevoted to the notion of algebraic hyperbolicity and its relationship with the main problem of this Thesis.
The third chapter contains a detailed discussion of known results for complements of sufficiently reducible plane curves.
In particular it describes the cases of complements of fourWhat is in this Thesisxilines and of a conic and two lines. For the latter it includes
a sketch of the proof as well as a description of a Theorem on
Greatest Common Divisor for S-units that is used in our extension of Corvaja and Zannier’s result. It ends with an overview of the generalization of previous theorems to ramified
covers of G2m obtained by the same authors.
The fourth chapter presents the details of our proof of the result for non-split complements of degree four and three components divisors. It starts with a description of the setting, together with its similarities and differences with respect to thesplit case. Then it introduces the study of the moduli space
for the divisors involved together with a suitable representation that is used in the proof. Finally a detailed proof with all
the computation ends the chapter as well as the second part.
The fifth chapter deals with a comprehensive discussion of
the methods present in literature for studying algebraic hyperbolicity of complements of plane curves in the algebraic
geometry setting. In particular it presents the ideas and a
sketch of the proof of Chen’s results enlightening its power
and its limitations.
The sixth chapter is an introduction to Logarithmic Geometry. It contains most of the definitions and the results thatare needed to discuss the theory of logarithmic stable maps.
Although not complete, it hopes to give the reader enoughelements to enjoy the constructions as well as the strength of
the results. The first section contains the basic backgroundon logarithmic structures, logarithmic schemes and maps, together with their basic properties. The second section discusses the notion of logarithmic stable curves and its relationship with the notion of usual stable curves. The last part describes in detail the theory of logarithmic stable maps, a generalization of Kontsevich construction, focusing in particular on
Chen and Abramovich-Chen theory of log stable maps to DF
pairs, which is the one used in the proof of the main theoremof the third part.

xii

Introduction

The seventh and last chapter contains the proof of algebraic
hyperbolicity for complements of very generic quartics. The
first section is devoted to the proof of an extension of Corvaja
and Zannier result for log-stable maps. Then, using this theorem, a reformulation of the problem using the moduli spaceof log-stable maps to DF pairs with suitable discrete data isgiven. Finally, using the properness of this stack, a deformation argument is described, showing that the stack is invariant
under flat deformation of the quartic, and applying the aforementioned moduli interpretation, obtaining the result fromthe three components case.

I.2

What is not in this Thesis

We end this introduction with a discussion of what has beenleft out of this Thesis.
The first thing that we should mention here, that should havebeen reserved a place in our presentation, is the complex analytic analogous of Conjecture 1 and its related open problems,such as the well-known Green-Griffiths Conjecture. Actually
the problem of degeneracy of integral curves in general type,or log-general type, manifolds and its relationship with the
problem of distribution of rational, or integral, points is oneof the starting point of Vojta’s analysis. One could say thatthe analogies between Nevanlinna Theory and value distribution Theory and Diophantine Approximation lies at the coreof Vojta’s Conjectures. Even deeper, Vojta built a dictionary
using which he was able to relate results in the complex analytic side with results in the Diophantine side and the otherway round, like Cartan’s Theorem in Nevanlinna Theory andRoth’s Theorem in Diophantine approximation. This allowed
him to set forth a series of striking conjectures which are only
briefly described in the first Chapter, and only from the Arithmetic point of view. We refer to [Voj1] and [Voj4] for moredetails.

What is not in this Thesis

xiii

Similarly all the recent results on Hyperbolicity problems andin particular on Kobayashi Conjecture are left out from ourdissertation. Since the work of Lang, a deep correlation between hyperbolicity in the sense of Kobayashi and distribution of rational and integral points on algebraic varieties has
been observed and studied. This link is even more strong
when one considers Diophantine problems over function
fields because in this case the focus on maps from smoothprojective curve into the variety has a lot of analogies withthe problem of describing holomorphic maps from the complex numbers to the corresponding analytic manifold. Thissimilarity allows one to try to adapt strategies and ideas coming from Complex Geometry in the function field case. Therefore a description of powerful techniques, such as Jet spaces
and Jet differentials, and the results obtained with these tools
could have been included. We decided not to include also theresults about the distribution of rational curves in surfaces
and higher dimensional varieties that have in common several aspects with the main problem of this Thesis. Among all
the works in this direction we cite [DSW], [SY], [Dem], [McQ],[NWY], [Rou] and [DMR], which are the most related to thecontents of this Thesis.
Looking at the arithmetic side of the problem considered, we
decided to focus on characteristic 0 arguments, not mentioning the problems arising in characteristic p. One of the reasons was that some of the results and the techniques cannot
simply be translated in this new setting. In particular some of
the Conjectures are simply false if merely stated in characteristic p. As an example there are unirational surfaces of general type in positive characteristic which provide counterexamples to Bombieri-Lang Conjecture (see [AV]). For similarreasons, when considering the function field version of Conjecture 1 we almost always assume that the base field is algebraically closed, although very interesting problems emerge
in the case one drops this assumption.
Finally, with the aim of keeping the exposition as self-containedxiv

Introduction

as possible, we tried to limit ourselves with the use of external theorems without a brief discussion on their statements.
In the cases where this has been done, we cited referenceswhere the reader can find detailed explanation of the results
used and of their proofs.

I
Lang-Vojta conjecture

1Lang-Vojta for Number
FieldsLang-Vojta Conjecture, named after Paul Vojta and Serge Lang,in its usual formulation reads as follows:
Conjecture 2 (Lang-Vojta). Let X be a quasi projective variety that can be written as Y \ D for a (smooth) projective variety Y
and an effective divisor D with normal crossing singularities, suchthat KY + D is big. Then any set of S-integral points on X is not
Zariski dense.
In this chapter we are going to review the history of this Conjecture, starting from his ancestor, namely the analogous results for algebraic curves, in particular Siegel’s Theorem on
integral points on affine curves and Mordell Conjecture, later
Faltings’ Theorem. Then we will move to the two dimensional case, which is the one we are more interested in in thisthesis: in particular, in the second section, we are going toreview the famous Bombieri-Lang Conjecture as the first attempt to extend Faltings’ Theorem to higher dimension, andto describe how Lang-Vojta Conjecture extends and generalizes it. Finally, in the last section we are going to describe newtechniques, originated by the new proof of Siegel’s Theorem
by Corvaja and Zannier in [CZ1], which helped to partially4

1. Lang-Vojta for Number Fields

answer Lang-Vojta Conjecture in some remarkable cases.

1.1

Integral points on Curves

The geometric classification of algebraic curves has been oneof the greatest mathematical achievements of 19th century.
This classification can be summarized in the following way:
every complete complex curve is birational to a unique nonsingular projective curve. To every curve is associated a discrete birational invariant, its genus. Finally, for a fixed genus g, isomorphism classes of projective non-singular curvesof genus g form a quasi-projective variety M g of dimension
max{3g − 3, g}. What can be said from the arithmetic pointof view? Here the problem is evidently far from being as accessible as the geometric classification. Actually the problem
of describe and characterize the distribution of rational andintegral points, can be considered to be completed only after
Faltings proof of Mordell Conjecture and includes major theorems by, among others, Dirichlet, Mordell, Siegel, Weil and
Faltings. Moreover there are still open problems regarding
arithmetic features of algebraic curves.
The difference sketched between the geometric and arithmetic
description of curves becomes more serious for surfaces: herea birational classification is available (although a complete description of moduli of general type surfaces is still lacking)whereas an arithmetic description of quasi-projective surfaces
has barely begun and lies mainly in deep conjectures like
Lang-Vojta’s one.
In this section we are going to review a sort of “arithmeticclassification” of curves through the most important featuresof Siegel and Faltings Theorems on the distribution of, respectively, integral points on affine curves and rational points on
projective curves. Our main motivation, together with thebeauty of these results, is to present the basis upon which the1.1. Integral points on Curves

5

higher dimensional analogous were formulated as well as toshow the interplay of different techniques which is a common
aspect of Diophantine Geometry.

1.1.1

Siegel’s Theorem

In [Sie] Siegel proved his celebrated Theorem on finiteness of
integral points on affine curves that reads as follows
Theorem 1.1.1 (Siegel 1929,[Sie]). Let C be an affine irreducible curve defined over a number field K and embedded in an
affine space Am . Let C˜ be the completion of its normalization and˜ its (geometric) genus. Let OS denote the ring of Sg = g(C)integers of K, for a finite set S of places of K containing the archimedean ones. Suppose that either g ≥ 1 or g = 0 and C˜ \ C contains
at least three points. Then C(OS ) is a finite set in Am (OS ).
Remark 1.1.2. Historically, Siegel’s original presentation of theresult was split in two parts: one dealing with the case of genus zero and three points at infinity and the remaining one
dealing with genus greater or equal than one. Moreover in
Siegel original formulation, the set S was the set containing
only the archimedean places of the number field K. The extension to number fields is due to Mahler in [Mah] for genus
1. The result has been finally extended to arbitrary finite set ofplaces by Lang in [Lan1] using an extension of Thue-SiegelRoth Theorem by Ridout in [Rid].
We introduce here the notion of Euler Characteristic of anaffine curve which will allow us to reformulate Theorem 1.1.1
in a convenient way that will be useful in the sequel.
Definition 1.1.3. Given an affine irreducible curve C , let C˜ be
the completion of its normalization and let S be the set of points
C˜ \ C . We define the Euler Characteristic of the curve C to be˜ = 2g(C)˜ − 2 + #S.
χ(C) := χS (C)Remark 1.1.4. Classically the Euler-Poincaré Characteristic of
a complete smooth curve, viewed as a compact Riemann sur-

6

1. Lang-Vojta for Number Fields

face, is equal to 2 − 2g. In particular what we have just defined coincide with the opposite of the topological definition
plus the number of points at infinity. Reasons for our choiceof sign will be given in the subsequent discussion of the notion of algebraic hyperbolicity in 2.2.2.
We can now rephrase Siegel’s Theorem 1.1.1 with the aid ofEuler Characteristic in the following way:
Theorem 1.1.5. Let C = C˜ \ T be an affine irreducible curve
defined over a number field K and embedded in an affine space Am .
Let OS = OSK denote the ring of S-integers of K, for a finite set S of
˜ >0places of K containing the archimedean ones. If χ(C) = χ T (C)mthen calC (OS ) is a finite set in A (OS ).
Sketch of the proof. We will suppose that S is the set of the archimedean places of K; with a few modification the proof canbe adapted to treat the general case.
1. Suppose we are given an infinite sequence of S-integral
points ( xi ) in C(OS ). There exists a convergent subsequence (with respect to an archimedean place) in C˜ with
˜ S ) ⊂ C(˜ Q), α is allimit α ∈ C˜. In particular, being C(Ogebraic.
2. If the genus is greater than zero we can embed C˜ inside
its jacobian Jac(C): this gives a sequence in the jacobian( xi0 ) which is convergent, up to replacing the sequence
by a subsequence. By weak Mordell-Weil Theorem theimages of all the elements of a subsequence in the group
Jac(C)/mJac(C) coincide. Hence we can assume thatxi0 = myi + z for some fixed rational point z ∈ Jac(C).
3. Define the map φ : Jac(C) → Jac(C) by φ( x ) = mx + z
and put D = φ−1 (C). The map is unramified and pullbacks of integral points are integral points. Hence φgives us a sequence yi of integral points that converges
to some β in D , eventually after passing to a subsequence.

1.1. Integral points on Curves

7

4. Next compare the height function on the two curves defined by mean of two pullbacks of the same very ample divisor on the jacobian (see Definition 1.2.12 for details on Heights). Using functoriality and the properties
of Neron-Tate heights on abelian varieties (in particular
the quadratic behavior with respect to the isogenies on
the jacobian coming from multiplication maps) we findthat the height on C grows like (a constant times) theheight on D to the power of m2 /2.
5. To conclude one applies Thue-Siegel-Roth in order tohave a lower bound for the height of the xi ’s on one
hand and uses the above described behavior of heights
together with the locally étaleness of the multiplicationby m map, for m large enough, to derive a contradiction. Roughly speaking the distance between points ofthe sequences and their limit does not change thanks tothe (local) étalness of the map φ; however Roth’s Theorem limits the velocity by which the sequence of theheights h( xi ) can converge while, on the other hand,the same speed on the cover D can be made arbitrarily big increasing m and this gives the contradiction to
the fact that infinitely many integral points can exists on
the curve C .
6. If the genus of the curve is zero, one is led to the caseof P1 minus, at least, three points. The conclusion inthis situation comes from an application of Roth’s Theorem. Equivalently one can reduce this case to the previous one in the following way: call Y this affine curve
whose completion has genus 0 and which posses at least
three distinct points at infinity. It can be shown (see forexample Theorem 5.1, ch 5, in [FWG+ ]) that there exists a cover Z of degree 3 of Y totally ramified over thethree distinct points in the boundary of Y. Therefore, by
Riemann-Hurwitz, Z is a curve of genus 1. Moreover
one can show that integral points of Y lifts to integral
points of the cover Z. The conclusion then follows from

8

1. Lang-Vojta for Number Fields

the already proven result for curves of positive genus.
Remark 1.1.6. The above proof of Theorems 1.1.1 and 1.1.5
uses Thue-Siegel-Roth Theorem which was unavailable at thetime Siegel formulated its Theorem; the original proof uses a
weaker statement, namely the the Thue-Siegel Theorem.
Moreover the language of jacobians was in some sense hidden in Siegel’s work which relies on properties of theta characteristics. A part from these differences both the original andthe modern proof follow the above described steps.
As can be seen by the previous sketch, a number of deep arithmetic and geometric theories (Heights machinery, MordellWeil and Thue-Siegel-Roth Theorems, the theory of Jacobians)
are used throughout the proof. This is going to be a fil rougecrossing most of the parts of this Thesis. Another proof ofthis Theorem by Corvaja and Zannier in [CZ1] will be given
in subsection 1.3.3 and will play a big role in the second part.
In the next subsection we are going to present Mordell Conjecture and Falting’s proof. This result, stating finiteness ofrational points in projective curves of genus greater than one,
implies and strengthen the cases of Siegel’s Theorem for thosecurves. Nevertheless, a part from genus zero and genus one
cases, height bounds obtained for integral points may be ofdifferent type with respect to the one given for rational points
coming from Faltings’ Theorem; hence Theorem 1.1.1 remains
of independent interest.

1.1.2

Mordell Conjecture

The history of Mordell Conjecture starts with the article[Mor1] by Mordell. In this seminal paper, after proving finite
generation for the group of rational points in an elliptic curve
defined over Q, he states one of the most famous problems
in Arithmetic Geometry, which has been proved by Faltings

1.1. Integral points on Curves

9

only sixty years after the original formulation. The Conjecture, now Faltings’ Theorem, reads as follows
Theorem 1.1.7 (Faltings [Fal1], Mordell Conjecture). Let
K be a number field and let C be a curve of genus greater than one
defined over K. Then C(K ) is a finite set.
This statement was highly non-trivial and only some particular case were known at Mordell’s time. Many mathematicians, although recognizing its power, were not convinced by
the conjectured result. André Weil commented
Nous sommes moins avancés à l’égard de la Conjecture de Mordell. Il s’agit là d’une question
qu’un arithmeticien ne peut guère manquer dese poser; on n’aperçoit d’ailleurs aucun motif
sérieux de parier pour ou contre. [Wei]We are less advanced in respect of the Mordell
Conjecture. This is a problem that every arithmetician can hardly not ask himself; nevertheless we do see no serious reason to bet for oragainst its truth.
In his general audience exposition of Mordell Conjecture andFalting’s ideas in [Blo], Spencer Bloch wrote «Probably mostmathematicians would have agreed with Weil (certainly I
would have) until [...] a German mathematician, Gerd Faltings, proved the Mordell Conjecture». This is even more revealing taking into account the proof by Grauert [Gra] andManin [Man] (although with a gap pointed out and correctedby Coleman [Col]) of the geometric case; see section 2.1 atpage 30 for further details on the function field case.
Nevertheless in 1983 Faltings presents a proof of Theorem
1.1.7 as a consequence of his proof of Tate Conjecture and Shafarevich Conjecture. His argument uses very refined and difficult tools like Arakelov Theory on moduli spaces, semistableabelian schemes and p-divisible groups. Vojta in [Voj3] (andpreviously for function fields in [Voj2]) gave another proof

10

1. Lang-Vojta for Number Fields

which uses ideas from classical Diophantine approximations
together with technical tools of intersection theory on arithmetic threefolds developed by Gillet and Soulè. After thisnew proof, Faltings in [Fal2] gave another simplification, eliminating the use of Riemann-Roch Theorem for arithmetic threefolds: using his new ideas he was able to extend previous results and to prove a Conjecture of Lang. Another simplification of both Vojta and Faltings’ proofs was given by Bombieriin [Bom] combining idea from Mumford [Mum] together withthe ones in the aforementioned papers.
A full proof of Theorem 1.1.7 goes beyond the scope of thischapter, we refer to the following books that contains detailed
and comprehensive discussion of the original proofs, together
with their subsequent simplifications: Bombieri and Gubler[BG] and Hindry and Silverman [HS] discuss Bombieri’s approach to Theorem 1.1.7. For an exposition of the ideas of Faltings’ original paper the main source is Faltings and Wüstholz
notes [FWG+ ]; another exhaustive treatment of the original
proof together with its link to Tate Conjecture and Shafarevich Conjecture can be found in Zarhin and Parshin article[ZP](1) .
The main importance of Faltings’ Theorem (together withSiegel’s Theorem 1.1.1) for the purpose of this Thesis is thefollowing corollary which completely describes the distribution of integral and rational points on algebraic curves:
Theorem 1.1.8 (Arithmetic classification of curves). Let C
be a projective, geometrically irreducible and non-singular curve
defined over a number field K. Let OS be the ring of integers of K
for a finite set of places containing the archimedean ones. Then, at
most after a finite extension of K(2) , the following holds:(1) For bibliographic history of this article we refer to the summary in thearticle’s arXiv page. We just note that the paper first appeared as an appendix to the Russian version of Lang’s book Fundamentals of Diophantine
Geometry.
(2) This assumption is made in order to give a unified treatment of thecase of genus zero and one. Indeed, at most after a finite extension, every1.2. Bombieri-Lang and Lang-Vojta Conjectures

11

Table 1.1: Arithmetic classification of curves
Genus
g=0
g=0
g=1
g=1
g≥2

Rational points
Infinite set
Infinite set
Fin. generated group
Fin. generated group
Finite set

Points at infinity
≤2
≥3
=0
≥1
Arbitrary

Integral points
infinite set
finite set
infinite set
finite set
finite set

Previous Theorem could also be restated using the Euler characteristic as defined in 1.1.3 in the same way done for Siegel’s
Theorem in 1.1.5. The main point of the previous classification is the fact that it exhibits a fundamental characteristicof Diophantine Geometry, namely the fact that, at least in aqualitative sense, Geometry determines Arithmetic. In particular
for curves the genus of the complex variety associated to thearithmetic curve determines the behavior of the distributionof rational points and, together with the number of points at
infinity, of integral points.

1.2

Bombieri-Lang and Lang-Vojta Conjectures

In the previous section we have seen how Siegel’s Theorem
1.1.1 and Faltings’ Theorem 1.1.7 completely describe the distribution of both integral and rational points on algebraic
curves over number fields. In this section we will describe
how these theorems for curves can, conjecturally, be extended
to surfaces. The idea is that, once a suitable geometric property that extends the role of the genus for curves is determined, analogous behavior of rational points could be established. However, as the geometry of surfaces compared to thealgebraic curve possess a rational point.

12

1. Lang-Vojta for Number Fields

geometry of curves is richer and subtler, the arithmetic of distribution of points in dimension two is far from being easy;
hence an elegant and concise description as given by Theorem 1.1.8 cannot be expected for arithmetic surfaces.
The correspondent statements for surfaces are contained in
the following two conjectures. The first one is due to Bombieriand Lang: Bombieri addressed the problem of degeneracy ofrational points in varieties of general type in a lecture at the
University of Chicago in 1980, while Lang gave more general
conjectures centered on the relationship between the distribution of rational points with hyperbolicity and Diophantine
approximation (see [Lan6] and [Lan2]). The conjecture reads
as follows:
Conjecture 3 ((Weak) Bombieri-Lang). Let X be a surface
of general type defined over a number field K. Then the set of Krational points of X is not Zariski dense.
While the former Conjecture can be seen as a two-dimensional
analogous of Faltings’ Theorem, Conjecture 2, which inspired
the title of this Thesis as well as most of its results, generalized and vastly extend Siegel’s Theorem on finiteness of integral points on curves with positive Euler characteristic. As
mentioned at the beginning of this chapter, this Conjecture isa reformulation of the (stronger) original Conjecture by Vojta
that uses ideas of Lang and admits the following analogous
statement which emphasizes the link with the Bombieri-Lang
Conjecture:
Conjecture 4 (Lang-Vojta II). Let X be a quasi projective variety of log-general type defined over a number field K and let OS
the ring of S-integers for a finite set of places of K containing thearchimedean ones. Then the set X (OS ) is not Zariski dense.
In this section we are going to describe how these two statements generalize the results of the previous section for curves,and how the geometric properties involved generalize theones used in Siegel and Faltings Theorems. We will then describe in the detail the two conjectures and mention some of1.2. Bombieri-Lang and Lang-Vojta Conjectures13

the known cases.

1.2.1

From Curves to Surfaces

From Theorem 1.1.7 one can see how the geometric properties encoded by the genus govern the arithmetic of the curve.
Seeking a generalization to higher dimensions, and in particular to surfaces, one is led to study which geometric features of the underlying complex variety determine the distribution of rational points. The first attempt would be to
study whether rational points on higher dimensional varieties are finite or not. However, parallel to this problem, one
can study a different characterization of the distribution of rational points: this follows from the fact that, up to a finite extension of the base field, each rational or elliptic curve (resp.
each affine curve with non positive Euler Characteristic) on
the variety will carry an infinite number of rational (resp. integral) points. Therefore it is natural to consider a weakerproperty rather than finiteness for rational points. As a motivation consider the following
Example 1.2.1 (Corvaja and Zannier, Turchet). Let X̃ be asmooth cubic surface defined over a number field K and let
H1 , H2 be two hyperplane sections such that H1 ∪ H2 consistsof 6 lines. Corvaja and Zannier in [CZ6] proved that the set of
S-integral points on X = X̃ \ ( H1 + H2 ) is not Zariski dense.
Moreover one can prove (see [Tur]) that the only families containing infinite integral points are the 21 remaining lines in X̃.
This example shows how in a complement of two completely
reducible hyperplane sections in a smooth cubic the S-integral
points are "almost finite" in the sense that, removing a finite
number of subvarieties (or a proper subvariety consisting of
the union of those), the S-integral points are finite in the surface. In particular the closure of the set of integral points is
a proper subvariety of the affine surface. Therefore an extension to higher dimensions of finiteness results for curves

14

1. Lang-Vojta for Number Fields

should look for non-density rather than to finiteness.
As pointed out in the arithmetic classification for curves, weshould allow finitely extension of the number field of definition of the surface X. This led to the following definition:
Definition 1.2.2 (Potential Density). Given an algebraic variety X defined over a number field K we say that the set of rational
points X (K ) is potentially dense if there exists a finite extension
F of K such that the set of F-rational points is Zariski dense in X.
In order to extend Mordell and Faltings ideas to surfaces wehave to look for geometric properties of algebraic surfaces
who could imply that the set of rational points is not potentially dense on the surface. Therefore we need geometric properties replacing, or better extending, the role played by thegenus in dimension 1. With this goal in mind we recall thefollowing
Definition 1.2.3 (Kodaira dimension). Let X be a smoothprojective algebraic variety and let KX be one if its canonical divisor.
For each m ≥ 1 such that the pluricanonical linear system |mKX |is not empty, i.e. such that h0 ( X, O(mKX )) 6= 0, let
ΦmKX : X → P N
be the associated map. The Kodaira dimension of X is defined tobe the number(−1
if h0 ( X, O(mKX )) = 0 ∀m
κ(X) =
max dim ΦmKX ( X ) otherwise .
Remark 1.2.4. 1. Some authors define κ ( X ) = −∞ insteadof −1 in the case in which all the pluricanonical linear
systems are empty. However we prefer the previous notation because is consistent with the analogous definition of κ as the transcendent degree of the pluricanonicalring of X minus 1.
2. From the fact that birational morphisms between varieties make their modules of differentials being isomor-

1.2. Bombieri-Lang and Lang-Vojta Conjectures

15

phic follows that the Kodaira dimension of a smoothprojective variety is a birational invariant.
3. It follows immediately from the definition that, for an
algebraic variety X, −1 ≤ κ ( X ) ≤ dim( X ).
We claim that the Kodaira dimension for curves gives an analogue classification like the genus one. First of all it’s easy
to show that for curves κ = min{1, g − 1} by a case by caseanalysis. Let C define a smooth projective curve.
• When the genus of C is zero the canonical divisor is
not effective because has degree deg KC = 2g − 2 < 0.
Hence κ (C) = −1 because all the pluricanonical linear
systems are empty.
• When the genus is one the canonical sheaf coincide withthe structure sheaf OC for which h0 (C , OC ) = 1 andh1 (C , OC ) = 0. Therefore κ (C) = 0.
• When the genus is greater than one the canonical divisor is ample and moreover for every m ≥ 3 the divisor mK is very ample (see for example [HM]) and henceκ (C) = dim C = 1.
Remark 1.2.5. It should be stressed that the classification of
curves by their Kodaira dimension reflects the complex analytic description of Riemann Surfaces in terms of curvature:
curves with negative Kodaira dimension, being of genus 0,
are Riemann Surfaces with positive curvature; elliptic curves,
i.e. curves with κ = 0 corresponds to flat Riemann Surfaces
and finally curves with positive Kodaira dimension corresponds to negative curved Riemann Surfaces, in particular to
hyperbolic ones.
From the previous list follows an arithmetic classification of
curves based on their Kodaira dimension. In particular thefollowing corollary of Faltings’ Theorem 1.1.7 holds:
Corollary 1.2.6. Given a smooth projective curve C definedover a number field K, the set of rational points C(K ) is not po-

16

1. Lang-Vojta for Number Fields

tentially dense if and only if κ (C) = 1 = dim C .
Motivated also by the previous result we recall the following terminology that extends the property of having genus
greater than one for curves.
Definition 1.2.7 (General type varieties). Let X be an algebraic variety. If κ ( X ) = dim X then X is said to be of generaltype.
For curves it is easy to see that most algebraic curves are ofgeneral type: in fact for almost all genus the curves are ofgeneral type and moreover the dimension of the moduli spacefor each genus is strictly increasing. Hence in dimension one“almost all” curves are of general type.

1.2.2

Bombieri-Lang Conjecture

The idea behind Lang and Bombieri conjectures about distribution of rational points on surfaces is that a rough analogousof the behavior exhibited by algebraic curves could hold also
for surfaces. First of all we recall the Kodaira classification of
surfaces which reads as follows
Theorem 1.2.8 (Kodaira Classification of Surfaces). Let X
be an algebraic smooth surface and let κ = κ ( X ) its Kodaira dimension. Then the following classification holds:• κ = −1: X is either a Rational or a Ruled surface.
• κ = 0: X belongs to one of the following four classes: Abelian,
hyperelliptic (or bi-elliptic), K3 or Enriques.
• κ = 1: X is an Elliptic Surface.
• κ = 2: by definition X is of General Type.
Recall that there are examples of elliptic surfaces of Kodairadimension strictly lesser than one: all Enriques surfaces andHyperelliptic surfaces are elliptic. However each surface with
κ = 1 is an elliptic surfaces. A finer classification together

1.2. Bombieri-Lang and Lang-Vojta Conjectures

17

with a complete description of each family of surfaces not ofgeneral type follows from the Enriques-Kodaira Classification
Theorem for surfaces (not necessarily algebraic).
We are interested in the behavior of the set of rational points
for each family of surfaces listed in Theorem 1.2.8. Let us look
more closer to each item appearing in the list:• Let us consider the first case: each Rational or Ruled
surface defined over a number field K is covered by rational curves which, by Theorem 1.1.7 have infinitely
many rational points. Therefore for all the surfaces inthis class the set of K-rational points is potentially dense.
• The case of null Kodaira dimension is more involved: it
is known that rational points are potentially dense forabelian varieties and for Enriques surfaces [BT1]. There
are several proved results of density of rational points
for some classes of K3 surfaces [BT3] and for Hyperelliptic surfaces [BT2]. Conjectures predicts that for eachof these classes the rational points are potentially Zariski
dense.
• For elliptic surfaces of Kodaira dimension one there isa gap for an arithmetic classification of surfaces based
solely on Kodaira dimension. In fact one can easily construct example of surfaces with κ = 1 that posses either a potentially dense set of rational points or a nondense one. Consider two fibrations X → C defined overa number field K having elliptic curves as fibers; suppose that the genus of the base curve C is greater thanone: then from Faltings’ Theorem 1.1.7 X has non (potentially) dense set of rational points. On the other hand
if C = P1 and there exists infinitely many sections over(a finite extension of) K, K-rational points are potentiallydense in X. In both cases it may happen that X has Kodaira dimension equal to one.
• For surface of general type it is expected that the set of

18

1. Lang-Vojta for Number Fields

rational points is not potentially dense: this has been
conjectured independently by Bombieri and Lang (whoconjectured a more detailed description of the set of rational points and extensions also to higher dimensions).
In particular, from previous discussion on Kodaira dimension for curves, it follows that Bombieri-Lang Conjecture implies Mordell Conjecture because curves of genus at least two are of general type.
Evidences for Bombieri-Lang Conjecture comes from the following Conjecture due to Lang and proved by Faltings in[Fal2] and [Fal3].
Theorem 1.2.9 (Lang Conjecture - Faltings’ Big Theorem).
Let A be an abelian variety over a number field K and let X be a geometrical irreducible closed subvariety of A which is not a translateof an abelian subvariety over K. Then X ∩ A(K ) is not Zariski
dense in X.
See [Hin1] for a detailed introduction and explanation of thisConjecture. From the previous Theorem it follows a corollary
which gives several evidences to Bombieri-Lang;Corollary 1.2.10. If X is a smooth projective variety of general
type defined over a number field contained in an abelian variety,then the set of rational points of X is not Zariski dense.
Following Noguchi’s proof [Nog] in the function field casefor varieties whose cotangent bundle is ample (which implies
that the variety is of general type) and using Faltings’ Big
Theorem (cfr. Theorem 1.2.9), Moriwaki in [Mor3] obtained
another evidence for Bombieri-Lang Conjecture which reads
as follows:
Theorem 1.2.11 (Moriwaki). Let X be a projective variety
over a number field K. If the sheaf of differentials Ω1X/K of X over K
is ample and generated by global sections, then the set of K-rational
points of X is finite.
Other evidences for Bombieri-Lang Conjecture comes from
related examples and conjectures for the distribution of ra-1.2. Bombieri-Lang and Lang-Vojta Conjectures

19

tional curves in general type surfaces, such as Bogomolov
Theorem on the finiteness of rational and elliptic curves on
general type surfaces with c21 > c2 [Bog].

1.2.3

Extensions and Lang-Vojta Conjecture

From the previous discussion we have seen that Conjecture 3
extends Faltings’ Theorem 1.1.7. It is natural to ask whether
a similar extension exists for Siegel’s Theorem 1.1.1. The answer is positive and it is related to Vojta’s ”landmark Ph.D.
Thesis", which gave the basis for a systematic treatment of
analogies between Nevanlinna Theory and Diophantine Geometry over number fields. Based on this analogy Vojta formulated a set of far-reaching conjectures. For a detailed description we refer to Vojta’s papers [Voj1] and [Voj4] as well
as chapters in the books [HS] and [BG].
We will now recall the basic definition needed to state themain Conjecture whose specific reformulation will give Conjecture 4.
Definition 1.2.12 (Weil’s Height Machinery). Let X be asmooth projective algebraic variety defined over a number field K.
There exists a (unique) map
h X,_ : Pic( X ) → { functions X (K ) → R}
well-defined up to bounded functions, i.e. mod O(1), whose image
h X,D for a class D ∈ Pic( X ) is called a Weil height associated toD. The map h X, satisfies:(a) the map D 7→ h X,D is an homomorphism mod O(1);(b) if X = Pn and H ∈ Pic(Pn ) is the class of some hyperplane inPn , then h X,H is the usual logarithmic height in the projective space;(c) - Functoriality - for each K-morphism f : X → Y of varieties
and for each D ∈ Pic(Y ) the following holds:
h X, f ∗ D = hY,D + O(1).

20

1. Lang-Vojta for Number Fields

By abuse of notation we will indicate for a divisor D ∈ Div( X ),the height corresponding to the class O( D ) ∈ Pic( X ), withh X,D . The previous definition can be extended to non smoothvarieties (even non irreducible ones) and over any field witha set of normalized absolute values which satisfy the productformula, see [Lan6] for further details. From the previous definition one can proof the following properties for the heightmachinery:
Proposition 1.2.13 ([HS],[Lan6]). With the above notation,
the function h X, satisfies:(d) Let D be an effective divisor in X then, up to bounded functions, h X,D ≥ O(1);(e) - Northcott’s Theorem Let A be an ample divisor in X withassociated height function h X,A then, for every constants C1 ,C2 and every extension K 0 of K with [K 0 : K ] ≤ C2 , thefollowing set is finite{ P ∈ X (K 0 ) : h X,A ( P) ≤ C1 }.
The second ingredient we need to introduce to formally state
Vojta Conjecture is the notion of local height. Morally wewant a function which measure the v-adic distance from apoint to a divisor D and such that a linear combination ofthis functions when v runs over the set of places gives a Weilheight for the divisor D. This motivates the following
Definition 1.2.14 (Local Height). Let X be a smooth projective variety defined over a number field K. Then there exists a map
λ_ : Pic( X ) → { functions

ä X \ suppD(Kv ) → R}v ∈ MK

defined up to MK -bounded function, i.e. up to constant maps Ov (1) :MK → R that are not zero for finitely many v ∈ MK , such that:(a) λ is additive up to MK bounded functions;1.2. Bombieri-Lang and Lang-Vojta Conjectures

21(b) given a rational function f on X with associated divisor div( f ) =D. Then
λ D,v ( P) = v( f ( P))up to Ov (1), for each v ∈ Mk where P ∈ U ⊂ X \ suppD (Kv )with U affine and max| P|v = 0 for all but finitely many v;(c) - Functoriality - for each K-morphism g : X → Y of varieties
and for each D ∈ Pic(Y ) the following holds:
λ g∗ D,v = λ D,v ◦ g + Ov (1);(d) if D is an effective divisor then λ D,v ≥ Ov (1);(e) if h D is a Weyl height for D thenh D ( P) =∑ dv λD,v ( P) + O(1)v ∈ MK

for all P ∈/ suppD, with dv = [Kv : Qv ]/[K : Q].
For detailed construction and related properties of local
height we refer to [Lan6] and [Ser]. One of the intuition behind the work of Vojta was the fact that local heights are arithmetic counterparts of proximity functions in Nevanlinna Theory: to see this consider a metrized line bundle L with a section s and metric |·|v : the function P 7→ log|s( P)|v is a local
height at v. Following Vojta [Voj1] one can introduce arithmetic proximity and counting functions for algebraic varieties
over number fields in the same spirit.
Definition 1.2.15. Let S be a finite set of places of K, and let
X, D as before. Then the following functions are well defined:
mS,D ( P) = ∑ dv λ D,v ( P)v∈S

NS,D ( P) = ∑ dv λ D,v ( P).
v/∈S

called the arithmetic proximity function and arithmetic counting function respectively. By definitionh D ( P) = NS,D ( P) + mS,D ( P).

22

1. Lang-Vojta for Number Fields

With this definitions we can now state the main Vojta Conjecture which translates Griffiths’ conjectural Second Main
Theorem in Nevanlinna Theory.
Conjecture 5 (Vojta). Let X be a smooth irreducible projective
variety defined over a number field K and let S be a finite set ofplaces of K. Let D be a normal crossing divisor, A an ample divisor
and KX a canonical divisor on X. Then for every e > 0 there exists
a proper closed subset Z such that, for all P ∈ X (K ) \ Z,
mS,D ( P) + hKX ( P) ≤ eh A ( P) + O(1).

(1.1)

We end this section by two easy propositions which show
how the above stated conjectured implies Bombieri-Lang Conjecture and Lang-Vojta Conjecture. We recall that one of theequivalent definition of bigness for divisor is the following: a
big divisor D has a positive multiple that can be written as thesum of an ample divisor B and an effective divisor E. In thefollowing proofs we will always assume that this multiple isthe divisor itself for simplifying the notation: this can be done
without loss of generality.
Proposition 1.2.16. Vojta Conjecture 5 implies Bombieri-Lang
Conjecture 3.
Proof. If X is of general type then KX is big, i.e. there exists a
positive integer n such that nKX = B + E with B ample and E
effective, and we will assume n = 1. Now Conjecture 5 with
D = 0 and A = B gives(1 − e ) h B ( P ) + h E ( P ) ≤ O (1).
By Proposition 1.2.13 h E ( P) ≥ 0 and hence, by Northcott’s
Theorem 1.2.13(e), the set X (K ) is not Zariski-dense in X.
In order to prove that Vojta Conjecture is stronger than LangVojta Conjecture we need the following reformulation of theproperty of being S-integral in terms of the functions definedin 1.2.15: a point P is S-integral if NS,D ( P) = O(1) and in particular mS,D ( P) = h D ( P) + O(1). We also recall the following:1.3. New techniques from Schmidt’s Subspace

23

Definition 1.2.17. Let X be a smooth projective variety andD a normal crossing divisor on X. X is said to be of logarithmic
general type, or log-general type, if KX + D is big for a canonicaldivisor KX of X.
Using this characterization of bigness cited before we can prove
the following
Proposition 1.2.18. Vojta Conjecture 5 implies Lang-Vojta
Conjecture 4.

Proof. For a log-general-type variety ( X, D ) one has
KX + D = B + E,
for B ample and E effective. Hence Vojta Conjecture with A =B gives for S-integral points(1 − e ) h B ( P ) + h E ( P ) ≤ O (1).
Thus, using Northcott’s Theorem, the set of S-integral pointsof ( X, D ) is not Zariski dense.

1.3

New techniques from Schmidt’s Subspace

In this last section of the chapter we are going to describe
Schmidt’s Subspace Theorem which vastly generalized Roth’s
Theorem. The importance of Schmidt’s results in this context
relies in a new proof of Siegel’s Theorem 1.1.1 due to Corvaja
and Zannier in [CZ1] and its implication to the study or integral points on surfaces. At the same time we will see how
Schmidt’s Theorem implies a particular case of Vojta Conjecture 5 for the complement in the projective space of a finite
union of hyperplanes in general position.

24

1.3.1

1. Lang-Vojta for Number Fields

Schmidt’s Subspace Theorem

Wolfgang Schmidt’s Theorem dealt with systems of inequalities in linear forms over Q with respect to one place. Schlickewei [Sch2] and Evertse [Eve] extended the results to arbitrary set of places on a number field with better estimates of
the quantities involved obtaining quantitative version of the
Theorem. A stronger formulation was given in [Voj1] by Vojta. For our purposes we present the following version basically due to Schlickewei in [Sch1]:
Theorem 1.3.1 (Schmidt’s Subspace Theorem). Let K be anumber field, S a finite set of places, e < 0. For every v ∈ S let
L0v , . . . , Lmv v be independent linear forms (in general position) in
X1 , . . . , Xn with coefficients in (an algebraic extension of) K. Thenthe set of projective solutions x ∈ PnK (K ) ofmv

| Liv ( x )|v< H ( x ) − n −1− e|x|vv ∈ S i =0∏∏

is contained in the union T1 ∪ · · · ∪ Th for finitely many hyperplanes T1 , . . . , Th in PnK . Here H (·) is the multiplicative projective
height H ( x1 : · · · : xd ) = ∏v max(| x1 |v , . . . , | xd |v ).
Remark 1.3.2. It can easily be seen that Theorem 1.3.1 extends
Roth’s Theorem: it is sufficient to consider the case for theinfinite place ∞ of Q in which m∞ = n = 2, L1 ( x, y) = x − αy,and L2 ( x, y) = y.
Theorem 1.3.1 has several application in Diophantine approximation of algebraic numbers which cannot fit in this shortanalysis of its statement; we refer the reader to detailed description in [BG] and [Bil].

1.3.2

Lang-Vojta for many components

Schmidt’s subspace Theorem can be used to prove Vojta Conjecture 5 for the complement of hyperplanes in general position in the projective space.

1.3. New techniques from Schmidt’s Subspace

25

Theorem 1.3.3. Conjecture 5 holds in Pn for D equal to thesum H1 + · · · + Hm where Hi is an hyperplane and { Hi } are in
general position.
Proof. Without loss of generality we can assume that each hyperplane is defined over the number field K, at most enlarging
K and considering a base change. Now the local height for Pn
with respect to the hyperplane Li is− log| Li ( x )|v,| x |v

where | x |v := max j | x j |v . Therefore the proximity function for
D can be written as
 | L ( x )| v
imS,D ( x ) = − ∑ ∑ log+ O (1).
| x |vv∈S i
On the other hand KPn = OPn (−n − 1) and hence
hKPn = −(n + 1)h + O(1).
Vojta Conjecture then has the following form:− ∑ ∑ logv∈S i

 | L ( x )| i

| x |vv− (n + 1)h( x ) ≤ eh( x ) + O(1),which is equivalent to (the logarithmic version of) Schmidt’s
Subspace Theorem.
In particular, as we are mostly interested in application to twodimensional varieties, an easy corollary is the following:
Corollary 1.3.4. Given 4 lines D1 , . . . , D4 in P2K defined overa number field K and S a finite set of places containing the archimedean ones, the set of S-integral points on the complement of
D1 + · · · + D4 is not Zariski dense.
Notice that the divisor formed by four lines in general position, makes the complement P2 \ D where D = D1 + . . . D4

26

1. Lang-Vojta for Number Fields

a variety of log-general type, because KP2 + D ∼ OP2 (1) isan ample divisor. In particular Lang-Vojta Conjecture 4 holds
for the complement of at least n + 2 hyperplanes in general
position in Pn .

1.3.3

A new proof of Siegel’s Theorem

In [CZ1] Corvaja and Zannier gave another proof of Theorem 1.1.1 avoiding the embedding in the jacobian and replacing the use of Roth’s Theorem in the original proof with thestronger Subspace Theorem (cfr. Theorem 1.3.1). The importance of this new reformulation, aside from simplifying
Siegel’s argument, relies on extension to higher dimension
which will be crucial in next parts of this Thesis.
Let us briefly summarize the ideas behind Corvaja and Zannier work. The statement we are going to recall is slightly
weaker than Theorem 1.1.1 nevertheless a covering argument
gives the full conclusion of Siegel’s Theorem as a corollary of
the following
Theorem 1.3.5 (Siegel, Corvaja and Zannier). Let C˜ be anirreducible projective smooth curve defined over a number field K
and let C be a non-empty affine subset of C˜ embedded in an affine
space Am . Assume that #(C˜ \ C) ≥ 3, then for a finite set of places
S containing the infinite ones, the set C(OS ) is finite.
We briefly sketch the proof of this Theorem in order to describe the main ideas.
Sketch of the proof. Let D be the divisor in C˜ whose irreducible
components are the points at infinity C˜ \ C and suppose that
D is supported on r points. For n  0 one has
h0 (C˜, nD ) = nr + O(1) =: l.
Let φ1 , . . . , φl be a basis for O( D ): we can assume that foreach S-integral point P, φj ( P) ∈ OS . This follows from the1.3. New techniques from Schmidt’s Subspace

27

fact that, up to multiplication by a non-zero constant, each φj
is integral over OS [ x1 , . . . , xn ], where x1 , . . . , xn are coordinate
functions on C . Assume now that there exist infinitely many
distinct integral points { Pi }. Up to passing to a subsequence,
we may assume that for each v ∈ S, the Pi converge v-adically˜ K v ).
to a point Qv ∈ C(
Estimate for |φj ( Pi )| are the following:(n|tv ( Pi )|−if Qv ∈ C˜ \ Cv ,kΦ( Pi )k 
1,otherwise
where Φ = (φ1 , . . . , φl ) and tv is the local parameter at Qv .
Taking into account the fact that φj ( Pi ) are S-integers, andputting S0 = {v ∈ MK : Qv ∈ C˜ \ C} one obtains thatn
H (Φ( Pi )) = ∏ kΦ( Pi )k  ∏ |tv ( Pi )|−
v .
v∈S

V ∈S0

The idea is now to consider elements z of O(nD ) with specified vanishing order at Qv : for this functions |z( Pi )|v becomes “small” as Pi approaches Qv . Now the vector spaceof z ∈ O(nD ) with prescribed order k of zero at Qv can bechoose to have positive dimension for specific values of k. Every z can moreover be written as a linear form in terms of theφi . Choosing appropriately vector spaces varying k gives rise
to linear independent linear forms in the φi which verify anequations as in Schmidt’s Subspace Theorem. Then the affine
version of Theorem 1.3.1 can be applied and one finds out that
only finitely many among the Pi are distinct which contradicts
the original assumption.
This new proof of Siegel’s Theorem has the advantage to besuitable for higher dimensional extension. In particular thesame authors obtained a number of strong results on degeneracy of integral points on surfaces by means of this new strategy. Among these results we cite the following which extends
Theorem 1.3.3 to more general situations.

28

1. Lang-Vojta for Number Fields

Theorem 1.3.6 ([CZ4]). Let X be a geometrically irreducible
nonsingular projective surface defined over a number field K and let
D1 , . . . , D4 be irreducible effective divisors such that
1. No three of them shares a common point;
2. For all i 6= j, suppDi ∩ suppD j 6= ∅;3. For all i 6= j, Di ∼ mD j for a certain m = mi,j ∈ Z.
num

Then no set of S-integral points in X \ D is Zariski dense.
Remark 1.3.7.
• Clearly for X = P2 all the hypotheses of
the Theorem are verified for the divisor D consisting offour lines in general position and hence the Theorem
implies Corollary 1.3.4. In particular each Di = Li iseffective (even ample in this case), the general position
hypothesis implies condition 1 and 2, while condition
3 follows from the fact that, being PicP2 of rank 1, allclasses of lines are linearly equivalent.
• The proof of the previous Theorem relies on a generalization of the ideas of Theorem 1.3.5 where a suitablechoice of a linear system of multiples of the irreducible
components Di replaces the linear spaces of rational
functions with prescribed order of zeros at limit points.
A more general theorem can be found in [CZ2] and, withsome modification and extension, in [CZ4]. It also worth mentioning a Corollary obtained by Levin in [Lev] where he wasable to drop the third hypothesis on the Di provided that Di
is ample for every i.

2Function Fields
Function fields in one variable and Number fields share several properties. This deep analogy was observed from thesecond half of the 19th century; one of the first systematic
treatment can be found in the famous paper by Dedekind andWeber [DW]. Further descriptions, due to Kronecker, Weiland van der Waerden, settled this profound connection which finally became formally completed with the scheme theory developed by Grothendieck. In this chapter we are goingto discuss how this analogy can be carried over Diophantine
problems. In the first section we describe the analogous of
Mordell Conjecture over function fields, which later became
Manin’s (although with a gap) and, independently, Grauert’s
Theorem in [Man] and [Gra] respectively. This serves bothas an introduction to arithmetic over function field and as astarting point for the generalization to higher dimension that
led to the formulation of Lang-Vojta Conjecture over function
fields. After recalling the notion of model which permits togive a unified definition for rational and integral points forvarieties defined both over number fields and over function
fields, in the last section we formulate Lang-Vojta Conjecture
for function fields, both in the split and the non split case, andwe connect it with the notion of algebraic hyperbolicity.

30

2.1

2. Function Fields

Geometric Mordell

In [Lan1] Serge Lang stated the following conjecture, which
has become known as Geometric Mordell Conjecture over C:
Conjecture 6 (Geometric Mordell). Let C be a curve of genus
greater than 1 defined over a (complex) function field K of an algebraic regular curve B , viewed as fibered surface π : C → B . If thegeneric fiber Ct has infinitely many rational points in K then there
exists a covering B 0 → B such that the base change C ×K K 0 , whereK 0 is the function field of B 0 , is isomorphic over K 0 to C0 ×C K 0 fora curve C0 defined over C. Moreover all but a finite number of these
sections arise from constant points of the fixed curve.
This Conjecture is obtained from Mordell Conjecture 1.1.7 replacing the number field with the a function field (in one variable). However a bare translation could not work as we aregoing to see in subsection2.1.2 and the isotriviality question
should be addressed. In order to properly describe the features of the previous Conjecture as well as the ideas behind
its proof by Grauert [Gra] we need first a precise extension ofarithmetic notions over number fields into the framework offunction field arithmetic.

2.1.1

Function fields and Number fields

As pointed out in the preface to this chapter function fields in
one variable behave very similarly to number fields. We start
this analogy with the following
Definition 2.1.1 (Function Field). A Function Field K overan algebraically closed field k is a finitely generated field extension offinite transcendence degree over k. A function field in one variable,or equivalently a function field of an algebraic curve, is a function
field with transcendence degree equal to one, i.e. a field K/k forwhich it exists an element x, transcendental over k, such that K is a
finite extension of k ( x ).

2.1. Geometric Mordell

31

Remark 2.1.2. With the language of schemes the function fieldof a curve X, or more general of every integral scheme overan algebraic closed field, can be recovered form the structure
sheaf O X in the following way: given any affine open subset of X the function field of X is the fraction field of O X (V ).
Moreover, if η is the (unique) generic point of X, the function
field of X is also isomorphic to the stalk O X,η .
The analogy between number fields and function fields ofcurves, also known as algebraic function fields in one variable, comes from the fact that one-dimensional affine integral
regular schemes are either smooth affine curves over a fieldk or open subset of the spectrum of the ring of integers of
a number field. Formally, given a number field k with ringof integers O the scheme Spec O is one dimensional affine
and integral. From this analogy, that was already present before the introduction of Grothendieck schemes’ theory, several classical properties of number fields find an analogue inthe theory of function field. In particular the theory of theheights can be defined over function fields.
Definition 2.1.3. Given a function field K in one variable of anon singular curve C , each (geometric) point P ∈ C determines a
non trivial absolute value by| f | P := e−ordP ( f ) .
Moreover if Q 6= P then the absolute values |·|Q and |·| P are notequivalent.
Remark 2.1.4.
• The definition could have been given
more generally for function fields of algebraic varieties
regular in codimension one (or rather for regular models of higher dimensional function fields), replacing
the point P with prime divisors. Extensions exist also
for function fields over non-algebraically closed fields
in which one should replace points with orbits under
the absolute Galois group.
• From the fact that any rational function f on a projec-

32

2. Function Fields

tive curve has an associated divisor of degree zero, it
follows that the set of absolute values satisfy the product formula.
Given the set of absolute values MK for a function field in
one variable K, normalized in such a way that they satisfy the
product formula, heights can be defined for K in the following
way:
Definition 2.1.5. Let K = K (C) be as before. For any f ∈ K
the height of f ish( f ) = − ∑ min{0, ordP ( f )} = ∑ max{0, ordP ( f )}.
P∈C

P∈C

In the same way for a point g ∈ Pn (K ), g = ( f 0 : · · · : f n ), its
height is defined ash( g) = − ∑ min{ordP ( f i )}.
P∈C

iFrom the definition it follows that a rational function on a regular curve has no poles if and only if its height is zero if andonly if it is constant on the curve.
We end this subsection with a table illustrating the interplay
and the similarity between number fields and function fields.
We stress in particular how each geometric object in the rightcolumn, in particular dominant maps and pullbacks, are analogous of purely arithmetic notions like extensions of fields
and extensions of ideals. This analogy can be brought further
using Arakelov Theory and extending the notion of divisors
to number fields by suitably compactifying the affine curveSpec OS ; in this framework an intersection theory can be defined for such compactified divisors sharing many analogous
properties of intersection theory in the geometric side. We
refer to [Lan5] for further details on this subject.

2.1. Geometric Mordell

33

Table 2.1: Number Fields and Function Fields analogy
Number FieldZ
Q
Qp
K finite extension of Q
place
finite set of places
ring of S-integers
Spec OK,S
product formulaextension of number fields
extension of ideals

2.1.2

Function Field
k[ x]
k( x)
k (( x ))K function field of C
geometric point
finite set of points
ring of regular functions
Affine curve C \ S
deg principal divisor = 0
dominant maps
pullback of divisors

Mordell Conjecture for function fields

Given the fact that many similarity between number and function fields exist, it is natural to ask if theorems and conjectures
about arithmetic (in a broader sense) still hold when replacing the occurrences of "number field" with "function field in
one variable" with proper modification according to Table 2.1.
Thus it seems reasonable to ask this question for Mordell Conjecture, as done by Lang. However, as the following example
shows, we cannot expect Faltings’ Theorem 1.1.7 to hold without a careful analysis of the new situation.
Example 2.1.6. Let C be a curve of genus greater than one defined over C and consider the trivial family C × P1 → P1 . The
family can be viewed as the curve C (trivially) defined overthe function field K (t) of P1 . All the fibers of the family, being
isomorphic to C have genus greater than one. Now Mordell
Conjecture over function fields, without any other restriction,
should imply that the set of C(t)-rational points of C , i.e. section P1 → C × P1 , are finite. However this is easily seen to34

2. Function Fields

be false by considering constant sections P1 → { P} × P1 foreach point P ∈ C . In particular the general type curve C defined over C (t) has infinitely many C(t)-rational points.
From the previous example one could guess that the problem
relied on the fact that the family was a product and the curve
C was actually defined over the base field C rather than on
the function field C(t), i.e. the family was trivial. However,as the following example shows, things can go wrong evenfor non trivial families.
Example 2.1.7 (Gasbarri [Gas]). Consider the curve C := ( x +ty)4 + y4 defined over C(t). It has an associated fibration
C → P1 whose generic fiber Ct0 = ( x + t0 y)4 + y4 is a smoothprojective curve of genus 3. Again if we consider the samestatement of 1.1.7 only replacing the number field with thefunction field C(t) we would expect that the number of C(t)rational points of C to be finite. However we claim that
C(C(t)) is infinite; to see this consider the equation α4 + β4 =1 over C2 : it has infinitely many solutions. Each solution
gives a C-point of Ct0 , namely (α − t0 β, β) proving the claim.
Moreover the family is not trivial in the sense of the previous
example, i.e. C is not defined over C. Notice however thateach fiber of the family is isomorphic to the curve x4 + y4 = 1
via x + ty 7→ x and y 7→ y.
Motivated by the previous examples we give the following:
Definition 2.1.8. Given a family of irreducible, smooth projective curves C → B over a smooth base B , we say that the family
is isotrivial if all the fibers Cb are isomorphic to a fixed curve C0 .
With abuse of notation we will say that a curve C defined over afunction field K is isotrivial if the corresponding fibration C → B
is isotrivial, where B is a curve with function field K.
Isotriviality extends the notion of (birational) triviality forfamily of curves, i.e. a product of curves fibered over oneof the factors is immediately isotrivial. At the same time thisnotion encompass many other families that are not products,
like the one defined in the previous example. However, after2.1. Geometric Mordell

35

a cover of the base of the family, each isotrivial family becomes trivial; in particular the following easy lemma holds:
Lemma 2.1.9. Given a isotrivial family C → B of smooth projective irreducible curves, there exists a cover B 0 → B such thatthe base changed family C ×B B 0 → B 0 with fibration given by second projection, is a trivial family, i.e. is isomorphic to a product
C ×C B0 .
The Lemma implies that rational points for curves definedover function fields will always be not finite for isotrivial
curves, at most after a base change. The analogous of Mordell
Conjecture for function fields thus asks whether this holds
only for this class of curves. We can then restate Conjecture 6
in the following way:
Conjecture 7. Let C be a smooth projective curve defined overa function field K of genus greater than 1. If C(K ) is infinite then C
is isotrivial.
The Conjecture has been proved in the sixties by Manin [Man](although with a gap fixed by Coleman [Col]) using analytic
arguments and later by Grauert [Gra] using algebraic methods. Samuel in [Sam1] gave a proof in characteristic p usingideas of Grauert. A detailed explanation of Grauert methods
can be found in Samuel’s survey [Sam2]. In Mazur’s detailed
discussion of Faltings’ proof of Mordell Conjecture [Maz] hestress the role of Arakelov [Ara] and Zahrin [Zar] results which imply new proofs of Geometric Mordell, i.e. Mordell Conjecture over function fields, using ideas of Parshin: this gives,
if necessary, even more importance to the geometric case.
One of the ideas of Grauert’s proof which is central in someof the higher dimensional extensions is the following: suppose C is a curve defined over a function field K of a curve
B , corresponding to a fibration π : X → B . Then one canprove that almost all sections of the fibration, which correspond to rational points, verify a first order differential equation, i.e. almost all sections are tangent to a given horizontal
vector field. Formally each section σ : B → X can be lifted to36

2. Function Fields

the projective bundle B → P(Ω1X ) = Proj(Sym(Ω1X )) via thesurjective map σ∗ Ω1X → Ω1B . Grauert proves (in a different
language) that there exists a section φ of a suitable line bundle over P(Ω1 ) whose zero section contains all but finitely
many images of sections. He then concludes that if infinitely
many sections exist, given the fact that they satisfy the differential equation given by φ = 0, a splitting is provided forthe relative tangent sequence which implies that the family isisotrivial (via the vanishing of the Kodaira-Spencer class).
In particular Grauert’s construction gives first insights for thetheory of Jet spaces which plays a role in some degeneracy
result for complex analytic analogues. In this direction recent analogue of Conjecture 7 in higher dimension have been
proved by Mourougane [Mou] for very general hypersurfaces
in the projective space of high enough degree using proper
extension of the ideas briefly described above.

2.2

Geometric Lang-Vojta

The previous discussion about similarity between number
fields and function fields can be brought further encompassing the notion of integral points. Moreover, another formulation of Vojta-Lang Conjecture can be given from a schemetheoretic point of view via the notion of model. In particular,
using insights coming from Conjecture 6 we can restate LangVojta Conjecture for number field case and its natural extensions to function fields in a unified way as follows.

2.2.1

Integral models and Lang-Vojta Conjecture

Let K be a number field, let OK denote the ring of integers of
K and let OS denote the ring of S-integers for a finite set ofplaces S containing the infinite ones. We give the following

2.2. Geometric Lang-Vojta

37

Definition 2.2.1. Let X be a smooth projective variety definedover the number field K. A (integral) model for X is a projective
variety X together with a flat morphism X → Spec OK such that
X ' Spec K ×Spec OK X .
Informally we can think X as a fibration over Spec OK whose central fiber over Spec K is (isomorphic to) X.
The definition of model can be generalized to deal with affine
varieties in a standard way. Moreover, if K is the function fieldof a smooth integral projective curve C , one can define a modelfor the variety X over K = k (C) replacing Spec OK with C . In
this way one can also see better the fibration interpretation
given before.
We are interested in describing rational and integral points forvarieties defined either over number fields or function fields.
For rational points, i.e. points of X (K ), the following factholds: let X be as before and fix an integral model X → B
with B = Spec OK for number fields, and B = C for function
fields, thenRational Points X (K )correspond bijectively
tosections π : B → X
A similar description holds for integral points. Let X̃ be asmooth projective variety and D ∈ Div(X̃ ) a divisor withnormal crossing singularities both defined over the number
field K; choose a model X̃ , D for X̃ and D over OS : in thesame way that rational points correspond to sections
π x : Spec (OS ) → X ,S-integral points (or rather D, S-integral points) correspond tosections such that π x does not intersect D over pointsv ∈ Spec (OS ) \ S, or equivalently such that π ∗ D is supported
over S. As before one can replace the number field K with

38

2. Function Fields

a function field of a smooth integral projective curve C andSpec (OS ) with C \ S for a finite set of points in C . Thus we
can reformulate this, denoting by B either Spec (OS ) or C \ S,saying that

(S, D)-integral points
corresponds bijectively tosections π : B → X such thatsupp(π ∗ D) ⊂ S.
Note that when the field K is replaced by a function field of
˜ (assume that the characteristic of the basea curve K = k (C)field k is zero, although this is not necessary for the statementof the Conjecture) a structure projection π : X̃ → C˜ is naturally given by the model, where the generic fiber Xt will beisomorphic to X = X̃ \ D. Nevertheless, as in Section 2.1,
a distinction should be made in order to deal with trivial or
isotrivial families. We gave the following definition for thetwo dimensional case:Definition 2.2.2. Let X be an affine surface embedded in an
algebraic projective smooth variety X̃ as X ∼= X̃ \ D for a normal
crossing divisor D, where X, X̃ and D are defined over a function
˜ corresponding to an affine fibration X → C . The
field K = k (C)fibration X → C is called split if X ∼= Xk ⊗k C over k, i.e. thefibration is trivial. In particular we call X split if the affine variety
X is defined over k.
Remark 2.2.3. One may argue that the previous definition defines birational trivial affine fibration which is certainly true
but we will see that a major role will be played by these fibrations in the sequel. In particular in the next section we aregoing to define a different version of Conjecture 2 for function
fields in which the main property will be independent by basechange of the original variety. In particular this would imply
that each isotrivial variety could be consider as birationallytrivial and hence “split".
For a good introduction to the theory of models over function

2.2. Geometric Lang-Vojta

39

fields and isotriviality problem see [Gas] sections 1 to 3.
The original formulation of Conjecture 2 for function fields
reads as follows:
Conjecture 8. Let X be an affine algebraic surface defined over
˜ for a smooth projective curve C˜ and sucha function field K = k (C)that the generic point Xt is of log-general type. If there exist sectionsof the structure projection π : X → C˜ of arbitrarily large heightand such that the union of their images is dense then X is isotrivial.
Notice the strong analogy with the previous statement andConjecture 7: in particular the case in which the X has dimension one, the divisor is empty, i.e. X is of general typeor equivalently of genus greater than one, is precisely Geometric Mordell. Therefore some evidences for Conjecture 8
comes for the proofs of Geometric Mordell as discussed insubsection 2.1.2. Moreover, by the previous unified description of rational and integral points over both number fields
and function fields, one can see how morphisms C˜ → X willplay an important role in the treatment of the previous Conjecture.

2.2.2

Algebraic Hyperbolicity

Conjecture 8 will not be the focus of the following parts of this
Thesis but rather another related formulation slightly weakerthat comes from the following observation made in the holomorphic curves setting. Recall that a complex compact manifold is said to be (Brody) hyperbolic if every holomorphic map
from C to the manifold is constant; in the compact case Brody
hyperbolicity is equivalent to Kobayashi hyperbolicity, equivalence following from Brody’s Lemma (see for details [Bro]).
In particular, for algebraic manifolds, hyperbolicity would
imply that all holomorphic maps from abelian varieties must
be constant. Kobayashi [Kob] and Lang [Lan4] made conjectures about the relationship between hyperbolicity and algebraic properties of the manifolds and whether such properties

40

2. Function Fields

would imply, or being equivalent to, hyperbolicity. One consequence of hyperbolicity for compact manifolds is the following:
Theorem 2.2.4 (Demailly [Dem]). Let X be a projective complex variety immersed in some projective space for a choice of a very
ample line bundle. Then if the associated manifold is hyperbolic thefollowing holds: there exists a constant A > 0 such that each irreducible curve C ⊂ X satisfies
˜ − 2) = Aχ(C)˜ ,deg C ≤ A(2g(C)where C˜ is the normalization of C .
Following this Theorem, Demailly introduced the following
notion which will be one of the focuses of this Thesis:
Definition 2.2.5. A smooth projective variety X is algebraically hyperbolic if there exists a constant A such that for each
irreducible curve C ⊂ X the following holds:
˜ .
deg C ≤ Aχ(C)Using strong analogies between hyperbolicity and degeneracy of rational points Lang conjectured that a general typevariety should be hyperbolic and in particular algebraically
hyperbolic. This has been extended to the relative case giving
the following
Conjecture 9. Given an affine variety X embedded as X̃ \ D
for a smooth projective variety X̃ and a normal crossing divisor D,
if X is of log-general type then there exists a proper subvariety Exc(called the exceptional set) such that there exists a bound for the
degree of images of non-constant morphisms C → X from affine
curves whose image is not entire contained in Exc, in terms of theEuler Characteristic of C .
Euler Characteristic for affine abstract curves was defined in
1.1.3 at page 5. We stress that Conjecture 9 is a special caseof Conjecture 8 because maps C → X from an affine curvewith normalization C˜ \ S correspond to (S, D )-integral points

2.2. Geometric Lang-Vojta

41

for X as defined over the function field of the curve C . Now
a bound for the degree of their images gives an height boundfor such integral points and therefore implies that they cannot be Zariski dense in X unless X is not isotrivial. Finally we
can now state the conjecture we are going to devote most of
this Thesis, namely a weaker version of Conjecture 9: the conjecture deals with logarithmic general type affine varieties inwhich there could exists curves with negative Euler characteristic. This implies that in general such varieties will not be
algebraically hyperbolic. Nevertheless a slight modificationof the bound of Theorem 2.2.4 it is expected to hold.
Conjecture 10. Given X, X̃, D as before, if X is of log general
type then there exists a constant A such that for each affine curve Cwith normalization C˜ \ S for a smooth projective curve C˜, a finite setof points S ⊂ C˜ and for all non-constant morphisms ϕ : C → X,
the following holds:
˜ ≤ A max{1, χS (C)}.
deg ϕ(C)Conjecture 10 is slightly weaker than Conjecture 9 because it
allows maps from curves of non-negative Euler Characteristic to X. However, if images of curves with negative Euler
characteristic are all contained in a proper subvariety of X,
e.g. there exist only finitely many images of such curves in X,
this implies the conclusion of Conjecture 9. For the property
described in the previous Conjecture, being a weaker versionof algebraic hyperbolicity, one can give the following
Definition 2.2.6 (Weak Algebraic Hyberbolicity). An affine
variety X embedded as X̃ \ D in a smooth projective variety X̃, fora normal crossing divisor D ∈ PicX̃, is called weakly algebraically hyperbolic if there exists a constant A such that for each
irreducible affine curve C ⊂ X the following holds
deg C˜ ≤ A max{1, χ(C)}.
From now on, when we refer to algebraic hyperbolicity wewill always considering weak algebraic hyperbolicity. When42

2. Function Fields

we will need to refer to the usual algebraic hyperbolicity wewill explicitly stress the difference, or we will call it strong
algebraic hyperbolicity.
Remark 2.2.7. Algebraic Hyperbolicity in its weaker form
seems to be the natural generalization of the projective caseto the affine case. If one considers the simplest case of complements of a normal crossing divisor D in P2 one sees thatthe smallest degree D can have in order for P2 \ D to be of loggeneral type is 4. However, each quartic possess 28 bi-tangent
lines (Plücker Theorem) and to each of these lines one can define a map
Gm → P2 \ Dgiven by the restriction of the map P1 → P2 whose image isa bi-tangent. Hence P2 \ D always admits maps from affine
curves with non positive Euler Characteristic and hence is
not strongly algebraically hyperbolic while it is expected to
be weakly algebraically hyperbolic. Nevertheless if these 28
maps, together with the 24 associated to the tangents to theflexes, are the only Gm immersion in P2 \ D this would not
violate Conjecture 9.

IIThree components case3
Known results
Before moving to our first original result of this Thesis weare going to review in detail what is known about Lang-Vojta
Conjecture for function field when the compactification of theaffine variety is the projective plane P2 . First of all we aregoing to describe the classical techniques available when thedivisor at infinity has several irreducible components focusing in particular on the case of the complement of four lines.
This presentation will allow us to see how the ideas of Corvaja and Zannier supersede the previous approach leading to
the solution for complements of a conic and two lines. We
will describe their approach and the greatest common divisor
formulation which will be essential in the next chapter. Finally we will briefly describe the ideas of the same authors
for a generalization to other type of surface which implies theresults for P2 .

3.1

Large number of components

From now on X will denote the variety P2κ \ D for an algebraic closed field κ of characteristic zero and a reduced normal crossing divisor D ∈ Div(P2 ). From the description givenin the preceding part X is of log-general type if and only if

46

3. Known results

deg D ≥ 4. Hence the split case of Lang-Vojta Conjecture forfunction fields (cfr. Conjecture 10) reads as follows:
Conjecture 11. Let D be a reduced projective plane curve withnormal crossing singularities and let C˜ be a smooth projective curve.
Let S be a finite set of points on C˜ and C = C˜ \ S the corresponding
affine curve. If D has degree greater than 3 then there exists a boundfor the images of non-constant morphisms φ : C → P2 \ D in termsof the Euler characteristic of C .
The normal crossing hypothesis here, as in all the versions ofLang-Vojta Conjecture stated before, is essential: we will see
in section 3.2 an explicit example due to Corvaja and Zannierof a divisor of degree four without normal crossing singularities where the conclusion of Conjecture 11 does not hold.
It turned out that the problem becomes more difficult the lessthe irreducible components of the divisor D are; as an example, no case of Conjecture 11 is known for a specific irreducible quartic (see the last part of the Thesis for results into
this direction). In this section we are going to review the proofof the Lang-Vojta Conjecture for complements of a completely
reducible quartic. This will serve as an example for Corvaja
and Zannier generalization to the three component case which will be the basis of our results presented in the next chapter.

Notations. We introduced a useful notation which we aregoing to use throughout this and the following chapter. For asmooth projective curve C˜ and a finite set of points S, in analogy with the number field case, we call S-integers the elementsof the ring of regular function in the affine curve C = C˜ \ S,i.e. elements of κ [C]. This functions will have all their polescontained in set S and their ring will be denote by OS . In
the same spirit, invertible elements of the ring of S-integers
will be called S-units: these are functions with zeros and polescontained in the set S. The group of S-units will be denotedby OS∗ .

3.1. Large number of components

3.1.1

47

The four line caseIn this subsection we are going to deal with the case in which
D has degree four and four irreducible components, i.e. D
consists of four lines in general position. We have alreadyseen a proof for number fields of degeneracy of S-integral
points, see Corollary 1.3.4 at page 25. The same argument
for function fields, mutatis mutandis, would give the corresponding result. However we will see how this results follows from an extension of Mason’s ABC Theorem for polynomials obtained by Brownawell and Masser [BM] and, independently by Voloch [Vol]. The theorem we will prove is thefollowing:
Theorem 3.1.1. Let D be the divisor of P2 consisting of fourlines in general position. Then for every smooth projective curve C˜and a finite set of points S ⊂ C˜ there exists a bound for the imageof non-constant morphisms ϕ : C → P2 \ D in terms of the Euler
characteristic of C , where C = C˜ \ S.
Proof. Given coordinates [ x0 : x1 : x2 ] for P2 , without loss ofgenerality we can assume that the equations of the four linesareD1 : x0 = 0

D2 : x1 = 0

D3 : x2 = 0

D3 : x1 + x2 = x0 .

Now morphisms C → P2 \ D corresponds to morphisms

C → A2 \ ( L1 + L2 + L3 ),where the affine lines have equations
L1 : x = 0

L2 : y = 0

L3 : x + y = 1,

Here the affine plane A2 is viewed as P2 \ D1 . Hence themorphism ϕ : C → A2 \ L, where L = L1 + L2 + L3 , is given
by P 7→ ( f ( P), g( P)) for a couple of regular functions on theaffine curve C , since ϕ avoids the line D1 . The fact that it48

3. Known results

avoids L1 and L2 implies that f and g are invertible elementsof κ [C], i.e. are S-units. In the same way the function 1 − f − gis an S-unit as well. Moreover the three S-units satisfy theequationf + g + (1 − f − g) = 1.
Thus the problem is equivalent to studying solutions in Sunits to the in-homogeneous equation
u1 + u2 + u3 = 1
and the result follows from the generalized S-unit Theorem ofBrownawell-Masser in [BM] (or equivalently using Voloch’s
result in [Vol]. Moreover, differently from the number field
case, for function fields there exists explicit bounds with effective constant for the height that, in this case, provided thatno subsum of f + g + (1 − f − g) vanishes, reads as follows:h( f , g, 1 − f − g) ≤ 3 max{0, 2g − 2 + #S}.

From the previous Theorem one can see how the geometricproperty of boundness of the degree for affine curves in theaffine surface P2 \ D has been translated into an arithmetic
property of a set of S-units. In particular methods dealing
with solutions of S-unit equations can be applied to recover
results related to the geometry of morphisms C → P2 \ D.
This will be a general principle in the subsequent generalization of Theorem 3.1.1 to the case of fewer components for thedivisor D.

3.2

The split case

Before moving to the case of three components we want to analyze the peculiar attributes of the previous Theorem related
to the general Conjecture 11 focusing on the field of definition of the algebraic varieties involved. In Theorem 3.1.1 the3.2. The split case49

divisor D has been chosen in a way such that the corresponding S-unit equation coming from its irreducible components
had a specific shape. This has been possible due to the factthat a degree four and four component divisor has no moduli, i.e. there exists a unique isomorphism class. This implies
the following corollary of Theorem 3.1.1:
Corollary 3.2.1. Let C˜, S, C , D as before. For every fibered
threefold X → C such that each fiber is isomorphic to P2 \ D there
exists a bound for the degree of sections C → X in terms of χS (C).
Proof. By the above discussion each fiber is isomorphic andhence the fibration X → C is isotrivial (independently of thebase C ). In particular by Lemma 2.1.9 at page 35 there existsa (finite) cover D → C such that the base changed fibration
XD := X ×C D is trivial. Then each section of the trivial fibration give rise to a map D → P2 \ D for which a boundof the degree is given by Theorem 3.1.1. The conclusion follows from the observation that height of k (D)-points is proportional to the height of corresponding k (C)-points via thedegree of the extension [k (D) : k (C)] and hence the corresponding degree for images of sections satisfies a similar relationship.
The difference between the settings of Corollary 3.2.1 and Theorem 3.1.1 is that for the former, the divisor D is assumed to
be defined over the function field k (C) whereas for the latter
D is defined over the base field C. Being D a divisor without moduli, at most after a cover D → C the two situations
coincide. This will not be true in general for log-general typesurfaces defined over function fields. However we can explicitly state a characterization of the so called split case.
Definition 3.2.2. Given a fibered threefold X → C where
C = C˜ \ S and the fiber X P ' X̃ \ D as before, with X̃ a nonsingular projective surface and D a divisor with normal crossing
singularities, we say that X is split if the field of definition of D
is C. This corresponds to fibration X → C where the fibers are50

3. Known results

all isomorphic, or equivalently, there exists a base change given bya covering D → C such that the fibration is trivial, i.e. that thefamily is isotrivial. Similarly the non-split situation corresponds
to a non isotrivial fibration or equivalently to the fact the field ofdefinition of D is not the base field.
As Corollary 3.2.1 showed, if the divisor D has no moduli,i.e. there exists only one isomorphism class for D, the split
case and the non-split case coincide. That will not be the casein general; in particular Conjecture 11 addresses the split casewhile a non-split case should deal with sections of non isotrivial fibration over a curve.

3.2.1

The conic and the two lines

Consider now the problem in which D is a union of a conic
D1 and two lines D2 , D3 : the normal crossing condition on D
is equivalent to the fact that both lines are not tangent to D1
and the points of intersection D1 ∩ D2 , D1 ∩ D3 and D2 ∩ D3
are distinct. For such divisors Conjecture 11 predicts a boundfor images of maps C → X = P2 \ D in terms of χS (C). Noticethat in this case D has moduli and hence the split case deals
with a different situation than the non-split one.
In [CZ5] Corvaja and Zannier solved this case of Conjecture
11 proving the following
Theorem 3.2.3 (Corvaja, Zannier). Let C˜ be a smooth complete projective curve, S ⊂ C˜ a finite set of points and D ⊂ P2
a degree four divisor consisting of a conic and two lines in general
position. Let f : C˜ → P2 be a non-constant morphism such that˜ verifies
f −1 ( D ) ⊂ S. Then the degree of the curve f (C)˜ ≤ 215 · 35 · max{1, χ(C˜ \ S)}.
deg( f (C))

(3.1)

Remark 3.2.4. 1. Morphisms f : C˜ → P2 such that f −1 ( D )is contained in S corresponds bijectively to morphisms
C → P2 \ D. In particular the Theorem implies Conjecture 11 for such divisors D.

3.2. The split case51

2. As mentioned at the beginning of this Chapter the normal crossing hypothesis is essential. In particular thesame authors proved that in the complement of a conic
and two lines, where the lines intersect in a point of theconic, there exists curves with vanishing Euler Characteristic and arbitrary large degree (this is Proposition 4.3
of [CZ5]). The proof is effective showing an explicit example of curves Gm → P2 \ D of degree n for every n.
Here is an outline of the strategy used in the proof:
Sketch of the proof. Giving a map f : C˜ → P2 such that f −1 ( D )is contained in S is equivalent to giving a map f : C → A2
where A2 is viewed as P2 \ D3 . Such a map is of the form
P 7→ ( x ( P), y( P)) for a choice of coordinates on A2 such thatthe affine equations for D1 and D2 are
D1 : y2 = x2 + λx + 1

D2 : x = 0,

for a constant λ (hence D3 is the line at infinity with respect
to this choice of coordinates). In the same way done for 3.1.1
the rational function x, y give rise to the following S-units:
u1 : = xu2 := y2 − x2 − λx − 1.

Similarly y ∈ OS being regular on the affine curve C . Moreover the two S-units and the S-integer verify the following
equationy2 = u21 + λu1 + u2 + 1.
The proof then proceeds by proving that each solution to theprevious equation has bounded height which gives the desired result by the fact that, for the map f ( P) = ( x ( P), y( P)),
one hasdeg f (C) ≤ HC˜ ( x ) + HC˜ (y).
Height boundness is proven by differentiate the equation by aspecific differential form without multiple poles and observing that the derived equation has many zeros in common withthe original one. The problem can be translated in a GCD

52

3. Known results

problem for (u − 1, v − 1) for S-units u, v and then a Theoremby the same authors gives the full conclusion.
We state here the detailed version of the lemma on the existence of a “proper derivation” on the curve C because we aregoing to need it in the next chapter.
Lemma 3.2.5 (Corvaja,Zannier [CZ5]). For every smooth pro˜ and every finite set of points S ⊂ C˜,
jective curve C˜ of genus g(C)there exists a differential form ω ∈ C˜ and a finite set T ⊂ C˜ of
˜ − 2} such that for every u ∈ O ∗cardinality ] T = max{0, 2g(C)S
there exists an (S ∪ T )-integer θu ∈ OS∪T having only simple poles
such thatd(u)= θu · ωu˜ .
HC˜ (θu ) ≤ χS (C)(3.2)

Moreover if a ∈ OS then there exists an a0 ∈ OS∪T such thatd( a) = a0 · ω˜ .
HC˜ ( a0 ) ≤ HC˜ ( a) + χS (C)˜ > 0 then there exists a regular
Sketch of the proof. If g = g(C)1˜ and hence one can define a 1-form ω withω̃ ∈ Ω = Ω (C)2g − 2 distinct simple zeros. Now for each S-unit u, the rational function θ = θu satisfying 3.2 has poles either in thezeros of ω or in S. Defining T as the set of zeros of ω makes θ
a S ∪ T-unit which obviously verifies˜ .
HC˜ (θu ) ≤ χS (C)If g = 0, given the fact that ]S ≥ 2, there exists a differential
form ω without zeros and with two distinct poles at two Spoints. Now by the same argument as before the set of polespf θ is at most ]S − 2 as wanted. The result for S-integers
follows the same ideas.
Example 3.2.6. As an example of application of the previous
Lemma consider the simpler equation
y2 = u1 + u2 + 1,

3.2. The split case53

with, as before, u1 , u2 ∈ OS∗ and y ∈ OS . Deriving with respect to the differential form given by Lemma 3.2.5 gives
2yy0 = u10 + u20 =u10
u0
u1 + 2 u2 .
u1
u2With some manipulation one gets the following two equalities:
u10 2
u0
y − 2yy0 = − 1 (w2 − 1)
u1
u1
u0
u20 2
y − 2yy0 = − 2 (w1 − 1),
u2
u2
where
 u0


u1
−
1
u1
u10
 u0
u 0  u1
w2 := u2 2 − 1 0 − 1.
u2
u1 u1

w1 : = u 1

1

−

u0

Hence by the fact that y divides both − u11 (w2 − 1) andu0− u22 (w2 − 1) we are led to consider the greatest common divisor between (w1 − 1) and (w2 − 1). Lemma 3.2.5 implies thatthe height of the functions w1 and w2 are bounded in terms ofS, T and g but not in terms of u1 and u2 , a feature which willbe critical in the subsequent applications.

3.2.2

Greatest Common Divisor estimates

From the sketch of the proof for Theorem 3.2.3 and the last example, one can see how algebraic hyperbolicity for P2 \ D canbe translated intro a problem related to divisibility inside thering of S-units. In this direction the main theorem by Corvaja
and Zannier reads as follows
Theorem 3.2.7 (Corvaja, Zannier [CZ5]). Let a, b ∈ OS∗ bemultiplicatively independent non-constant S-units, and let α, β be54

3. Known results

positive integers. Then one of the following condition holds:•

H ( a) ≤ α[k (C˜ : k ( a, b)]andH (b) ≤ β[k (C˜ : k ( a, b)]•∑ min{v(1 − a), v(1 − b)} ≤(α + 2β) H (b) + βH ( a)αβ + α + β

v/
∈S

+

αβ + α + β − 1˜ .
χS (C)
2

Remark 3.2.8. Theorem 3.2.7 can be seen as a result on thegreatest common divisor between 1 − a and 1 − b, or better a
S-gcd, namely the number of common zeros of the two rational functions outside the set S. In particular an explicit
choice for the constants α, β can give an upper bound which
depends only on the height of the S-units as well as on the
Euler Characteristic of the underline curve C˜.
Theorem 3.2.9 (Corvaja and Zannier). Let a, b ∈ OS∗ notboth constant, and let H := max{ H ( a), H (b)}. Then(i) If a, b are multiplicatively independent, we have√3

∑ min{v(1 − a), v(1 − b)} ≤ 3 2( H (a) H (b)χ(C))

v/
∈S

√
1
3
≤ 3 2( H 2 χ(C)) 3 .

1
3

(3.3)

(ii) If a, b are multiplicatively dependent, let ar = µbs be a generating relation. Then either µ 6= 1 and∑ min{v(1 − a), v(1 − b)} = 0,v/
∈Sor µ = 1 and∑ min{v(1 − a), v(1 − b)} ≤ min

v/
∈S

≤



H ( a) H (b)
,
|s|
|r |

H
.
max{|r |, |s|}



(3.4)

3.3. Extension to ramified covers of G2m

55

Recall that a generating relation for two multiplicative dependent elements is a relation ar = λbs such that the pair (r, s)
generates the lattice of all the pairs (ρ, σ) such that aρ = λbσ .
This result is the function field analogue of a theorem by thesame authors obtained in the arithmetic case (see [CZ5]).
We end this section mentioning a link between these resultsand Lang-Vojta Conjecture: as pointed out by Silverman in[Sil] this link is profound and goes beyond the applications
to function fields arithmetic. For this let us briefly recall thedefinition of generalized (logarithmic) greatest common divisor ina number field:
Definition 3.2.10 (Silverman [Sil], Corvaja and Zannier[CZ3]). Given a, b in a number field k, the greatest common divisor of a, b is defined to be
hgcd ( a, b) = ∑ min{max{0, v( a)}, max{0, v(b)}},where the sum runs over all the places of k. An S-gcd will be definedin the same way by summing over all the places outside the finite
set S.
Silverman noticed that the former is actually a Weil Heightin the blow-up of P1 × P1 over (0, 0) with respect to the exceptional divisor. From this observation one can prove that
Vojta Conjecture 5 for such a blow-up (with respect to a suitable ample divisor and with some natural hypotheses) implies an upper bound for the greatest common divisor of acouple of algebraic points (this is Theorem 6 of [Sil]). Therefore, in some sense, a gcd argument applied to Lang-Vojta
Conjecture is very natural.

3.3

Extension to ramified covers of G2m

The fundamental characteristic of divisors D in Theorem 3.2.3
was the presence of three irreducible components: in particular the three components implies the existence of S-units and56

3. Known results

an S-integer satisfying an equation whose solutions (and theirheights) lie at the core of Corvaja and Zannier methods. Thus
it seems natural to ask if similar results could be obtainedfor complements of three component divisors in more general surfaces other than P2 . It turns out that is true and is
the content of a recent generalization of Theorem 3.2.3 by thesame authors obtained in [CZ7] which we are going to recall
because of its implications with the results of the next chapter.
The starting point is the following easy observation:
Remark 3.3.1. Let D = D1 + D2 + D3 be the divisor in P2
formed by a conic and two lines in general position. Let f i = 0
be the equation of the i-th irreducible component assuming
deg D1 = 2. Then the following map P2 \ D → G2m to thebi-dimensional torus is dominant:
P 7→

f

f2
1
( P ), ( P )
2
f3
f3



.

It turns out that the map defined above is the only feature
required for algebraic hyperbolicity of some log-general typesurface.
Theorem 3.3.2 (Corvaja and Zannier [CZ7]). Let X be asmooth affine surface with a finite map π : X → G2m such thatthe closure of the image of the ramification divisor Z does not passthrough the singular points of the boundary of G2m . If X is of loggeneral type then there exists a constant γ = γ( X, π ) such thateach affine curve C ⊂ X verifies:
deg C ≤ γ · max{1, χC }.
From the previous Theorem and the remark follows a result
for complements of P2 that generalizes Theorem 3.2.3 encompassing more general cases. The result reads as follows
Theorem 3.3.3 (Corvaja and Zannier). Let D a plane curvewith normal crossing singularities of degree at least four and at

3.3. Extension to ramified covers of G2m

57

least three irreducible components. Then there exists a constantδ = δ( D ) such that each affine curve C in P2 \ D verifies
deg C ≤ δ max{1, χC }.
Clearly this result implies Theorem 3.2.3 and Theorem 3.1.1
although with possibly different constants involved.

3.3.1

Extra divisor from ramification

The proof of Theorem 3.3.2 and hence of Theorem 3.3.3 relies
on an estimate of the contribution of the ramification divisor
to the degree of the affine curves in the surfaces. The ideas
behind the proof are the following:
1. The starting point is that for each map C → X the contribution of the ramification divisor to its degree is small.
More precisely let ϕ : C → X be a non-constant map
from an affine curve C to the affine surface X, such thatits image is not contained in the ramification divisor
Z ⊂ X of the finite map π : X → G2m . Then for every e > 0 there exists a constant Γ (which depends on
X, π and the Euler characteristic of C ) such that for every such ϕ either the height H = H ( ϕ) of ϕ verifies
H ≤ C or the pullback divisor ϕ∗ ( Z ) is such thatdeg ϕ∗ ( Z ) ≤ eH.
The bound follows from a height bound for zeros ofpolynomials without repeated factors obtained in [CZ5].
2. From the fact that the affine surface X is of log-general
type one can prove that the ramification divisor is big,
being linearly equivalent to KX̃ + π ∗ ( E) where E is thedivisor P2 \ G2m , whose bigness is equivalent to X bea surface of log-general type. Here π denotes also the“completed” map π : X̃ → P2 and E is viewed as adivisor on P2 .

58

3. Known results

Being big, Z is in the closure of the effective cone of X
and hence it’s intersection with every curve C˜ in X isbounded below by a constant times the degree of C˜ (apart from a finite set of curves).
˜ with the (com3. Now the intersection of the image ϕ(C)pletion of) the ramification divisor Z̃ is bounded frombelow by the previous argument and at the same time
its affine part is bounded from above. This implies thatthe most part of the intersection should be concentrated
on the points at infinity, i.e. on S. This can be reformulated in terms of height bounds for S-units solutions
to a sum of monomials with non zero constant term andtherefore follows under the cases of Brownawell-Masser
generalized S-units Theorem. In particular this implies
that either the S-units, and hence ϕ(C), have boundedheight or they verify a dependence relation which gives,
again, a bound for the degree of the immersed curve.
The previous sketch of the proof shows that the result has
been reduced to an application to Brownawell-Masser Theorem as in the four line case: in particular the ramification
divisor Z plays the role of the missing fourth component of
D that appears in Theorem 3.1.1. It is expected that similar
results holds in the non-split situation using ideas similar tothe one exposed in the next chapter.

4
The non-split case
4.1

Statement of the problem

The main goal of this chapter is to generalize the situation ofchapter 3 to the so-called non-split case, i.e. the case of LangVojta Conjecture for the complement of a conic and two lines
in P2 , where now the divisor is defined over the function fieldof a curve rather than on C. As in the constant case we aregoing to reduce the problem to solve an equation and boundthe height of its solutions with Corvaja and Zannier method.
In our case the equation that describes this setting reads asfollows:
y2 = u21 + λ( P)u1 + u2 + 1.

(4.1)

Here again y is a S-integer and u1 , u2 are S-units. We note that
this equation is precisely the same considered in [CZ5] where
the polynomial in the right-hand side has now non constantcoefficients. Geometrically this corresponds to the data of an(affine) threefold X fibered over the curve C where each fiber
is isomorphic to P2 \ D and D is a divisor consisting of a conic
and two lines. Each solution of the equation (4.1) gives a section of the fibration X → C .

60

4. The non-split case

The situation considered in this chapter is made explicit in thefollowing diagram:
AX
σ



C

(4.2)

π
λ

/ P1

Here the parameter λ( P) is a rational function of the crossratio of the four singular points on the conic of the divisor
on the fiber over P and σ is a section of the projection π (see
4.3 for a detailed description of the geometric setting). We
observe that this is the natural generalization of the settings
considered in chapter 3: morphisms from the affine curve
C = C˜ \ S to P2 \ D can be seen as sections of the trivial(P2 \ D)-bundle over the curve C . In the case considered inthis chapter the trivial bundle is replaced by a fibration inwhich the divisor at infinity is moving. Moreover, generalizing the constant case, the three irreducible components ofthe divisor D = DP are not supposed to be in general position for every P ∈ C˜ (although we need some restrictions on
the “degeneracy” of the divisor).
The main result of this chapter is the following
Theorem 4.1.1. Let C˜, S, X as above. Let σ : C → X be a nonconstant section for the fibration π : X → C where each fiber isisomorphic to P2 \ D. Then, in a suitable projective embedding of
the variety X, if the fibration is not birationally trivial the degree of˜ verifies
the curve σ(C)˜ ≤ 213 · 58 · χS (C)˜ + 28H ˜ (λ) + 8H ˜ (λ).
deg(σ(C))
C
C
Example 4.1.2. Consider a plane smooth curve C in P2C . For
each point P ∈ C let t P denote the tangent line to C at P. Thisdefines a fibration over C in the following way: over a pointP ∈ C let X P be the complement in P2 of the divisor formed bya fixed quadric D1 , the line at infinity D∞ (assuming a choice4.1. Statement of the problem

61of coordinate has been made) and the line D3,P = t P . A picture of this situation can be seen in Figure 4.1.
Figure 4.1: Fibered threefold

D3, P1

D3, P3

D3, P2

X

D1

D1
D¥

D1
D¥

-1

D¥

-1

Π HP1 L-1

Π HP2 L

Π HP3 L

Π

P2
P3

P1

C

The threefold

X =[XP → CP∈C

can be seen as a surface defined over the function field of thecompletion of the normalization C˜ of C where a point of the
˜ corresponds to a section σP : C˜ → X̃
surface P ∈ X (k(C))−1such that σP ( D ) ⊂ C˜ \ C . In particular in the case in which
the divisor[
D=
D1 + D∞ + D3,P
P∈C

has normal crossing, being deg D = 4, each fiber is of loggeneral type and hence Theorem 4.1.1 can be applied giving
˜ as expected by Cona bound for the degree of images σP (C)jecture 10.

62

4.2

4. The non-split case

Moduli of three components divisors

In this section we will analyze configurations of a conic andtwo lines in P2 . Our aim is to prove that a moduli space for
equivalence classes of these divisors is of dimension one.

P1

P4

P2

D2

P3
D1

D3

Figure 4.2: Configuration of a conic and two lines in general
position
For this let D be the sum of a conic D1 and D2 , D3 distinct lines
in P2 defined over κ. This divisor has four singular points,
the four points of intersection between D1 and D2 + D3 ; thesepoints are distinct in the case when D1 , D2 , D3 are in general
position, i.e. D has normal crossing singularities. We want tocharacterize completely isomorphism classes of such divisors.
First we observe that, applying birational automorphisms of
P2 each class possesses a representative with a fixed conic D1
as component of degree two. Hence the problem can be reduced to study isomorphism classes of couple of lines nottangent to D1 whose intersection is not on the conic. Oneof such divisor is visible in figure 4.2. Secondly one can notice that the problem is equivalent to the study of fourplesof points on P1 , via the isomorphism between the conic andP1 , that give rise to isomorphic divisors (here we take theline D2 as the one passing through the first two points, andthe line D3 passing through the last two points). In other4.2. Moduli of three components divisors

63

words a moduli space for our problem will be represented
by a scheme with a map from M0,4 ∼= Gm \ {1}, where thelast isomorphism is given by the cross-ratio. However, although the cross-ratio of the four points gives information
about the divisor it does not characterize completely an isomorphism class. As an example consider the following two
fourples ( P1 , P2 , P3 , P4 ) ( P2 , P1 , P3 , P4 ): clearly they define thesame divisor but the two cross-ratios are inverse of each other.
Hence configurations of four points with the same cross-ratio
give rise to isomorphic divisors, but the converse is not trueas shown by the preceding example. However the following
basic lemma holds:
Lemma 4.2.1. Given two fourples P = ( P1 , . . . , P4 ) andQ = ( Q1 , . . . , Q4 ) of points in P1 . At most after applying themap mapping the first two points of each four-ple to 0, ∞ and applying the basic theory of cross-ratio, we can assume P3 = Q3 andP4 = Q4 . If neither the two fourples have the same cross-ratio nor
Q is obtained by P by permuting the points, then the configurationof divisors defined by P and Q are not isomorphic.
Hence we are reduced to calculate which permutations of four
points give rise to isomorphic configurations of divisors. We
can then consider the action of the permutations’ group S4 onan ordered set of four points in the projective line, i.e. an element of (P1 )4 ; an easy case by case analysis shows that the
subgroup of S4 that leaves the divisor unchanged is
G = h(12), (13)(24), (14)(23)i. Hence, by classic properties of
the cross-ratio, the only generator of G that changes the crossratio and under which the divisor configuration is invariantis (12). Thus in order to completely describe isomorphismclasses of degree four and three components divisors in P2 it
is sufficient to define a map()
degree four and three
0
λ :
−→ P1
components divisor in P2
constant on isomorphic divisors. By the description given

64

4. The non-split caseabove we obtain a natural 2:1 map, from the moduli space
M0,4 to the moduli space of degree four and three components divisors: this map simply associates to each fourple thetwo lines passing to the four points. With abuse of notation
we indicate as λ0 the composition(
λ0 : M0,4 →degree four and threecomponents divisor in P2)→ P1 ,

which is defined as
β ( P1 , P2 , P3 , P4 )2 + 1−2
β ( P1 , P2 , P3 , P4 )2
β( P1 , P2 , P3 , P4 ) − 1=,
β( P1 , P2 , P3 , P4 )λ0 ( P1 , P2 , P3 , P4 ) =(4.3)

where β is the cross-ratio. From this definition λ0 is a morphism from the (ordered) quadruple of points in the conic
D1 which associates to every configuration of the divisor D
a point of P1 . However λ0 is a function of the cross-ratio of
the quadruple P1 , . . . , P4 and so it is defined only when there
are at least three distinct points. In our situation, requiringthat over the affine curve C the fiber is P2 \ D and D has four
singular distinct points is equivalent to require that the set S
contains all the poles and zeros of β and hence all the polesof λ0 : this implies that some cases of non general position are
allowed but only over points in S. We will moreover enlarge
S such that it contains all the zeros of λ0 : this assumptions ismade in order to include the case of cross-ratio 2 (given by
the factor -2 in the preceding formula) where the quartic has
a non-normal crossing singularity, and also to apply Theorem 4.4.6 and has the advantage that there will be no need of
distinguish between the case of negative and positive Euler
characteristic. At the same time, this is not a strong restriction because λ0 will be a datum of the variety we want to deal

4.3. Affine threefolds

65

with and hence it does not depend on the method used forthe proof.
With abuse of notation we will sometimes indicate the value
λ0 ( P1 , . . . , P4 ) as λ0 ( D ) where the configuration of D is defined by the points P1 , . . . , P4 on the conic D1 .

4.3

Affine threefolds

We are interested in a specific class of affine threefolds fiberedover affine curves which generalizes the trivial P2 \ D-bundle
considered in the split case. More in detail we consider thefollowing class of affine threefolds:(?) X is an affine threefold fibered over the affine
curve C such that the completion of the fibration is
the trivial P2 -bundle over C˜. Every fiber π −1 ( P)for a point P ∈ C˜ is of the form P2 \ DP where
DP is a divisor of P2 of degree four formed by anirreducible conic and two lines such that there areat least three distinct singular points. If the point
P is in C then the function λ0 is regular on DP .
(see the diagram 4.2) . As an example of this situation one canconsider the bundle π : X → C where C˜ is a plane rational
curve and the divisor DP is formed by the conic x2 + y2 = 1,
the line at infinity (in affine coordinates) and the line t P , i.e.
the tangent line to the curve at the point P. In this example S
will contain every point P ∈ C˜ such that DP is a pole for λ0 .
It follows from the definition of the class (?) that giving sucha threefold is equivalent to giving a rational function
λ : C˜ 99K P1 ,which associates to a point P ∈ C a point of P1 viewed as thevalue of the function λ0 ( DP ), i.e. λ( P) specifies the isomorphism class of the divisor DP in the fiber over P. More in detail the affine threefold will be determined by the properties

66

4. The non-split caseof the divisor D = ∪ DP which can be described as follows:
we can naturally embed X inside X̃ := C˜ × P2 and denote by
p1 : X̃ → C˜ and p2 : X̃ → P2 the two projections. Then the
fibration X → C is uniquely determined by a line bundle µ ∈˜ and the choice of a divisor D ∈ | p∗ (µ) ⊗ p∗ (OP2 (4))|,
Pic(C)
2
1
see diagram below.
p1∗ (µ) ⊗ p2∗ (OP2 (4))

X
A
σ




/ C˜ × P2

π

C

 




p2

(4.4)
/ P2

p1

/ C˜

However not every divisor in the linear system gives rise toa fibration satisfying condition (?): we impose the condition
that D |π −1 ( P) has three components. This in particular implies
that on every fiber the divisor is determined by the value ofa function of the map λ0 defined above, i.e. over every pointP ∈ C the fiber is uniquely specified by the value of λ0 on the
singular points of DP , which we assume to be at least three,and such that the cross-ratio has no pole for this configuration.
In particular we will prove that every threefold satisfying (?),
described by a non constant rational map λ : C˜ 99K P1 , has
images of sections with bounded degree in terms of the Euler
Characteristic of the base curve.

4.4

Proof of the result

From now on we will work on an affine algebraic variety ofdimension three satisfying condition (?). We will denote by
DP the divisor defined on the fiber over the point P (or sometimes just D where the point we refer to is clear) and its three4.4. Proof of the result

67

irreducible components will be indicated by D1 (the conic)
and D2 , D3 (the two lines). The function λ : C˜ → P1 will
denote the map defined by
P 7 → λ 0 ( D P ).
We will suppose, at most after enlarging S, that λ is a S-unit,
i.e. S contains all its zeros and poles. We begin by proving thefollowing:
Lemma 4.4.1. Let C˜, S be as before and let π : X → C be anaffine fibered threefold verifying condition (?) and characterized bya non-constant rational function λ. Let σ : C → X be a section ofπ. Then there exist S-units u1 , u2 ∈ OS∗ and an S-integer y ∈ OS
satisfying
y2 = u21 + λu1 + u2 + 1(4.5)

and such that deg σ(C) ≤ HC˜ (u1 ) + HC˜ (y).
Proof. From condition (?) it follows that, after a choice of homogeneous coordinates, we can consider affine coordinates( x, y) in every fiber with respect to the line D2 viewed as theline at infinity x0 = 0. In this system of coordinates, without
loss of generality, the line D3 has equation x = 0 and the conic
D1 has equation y2 = x2 + λx + 1. Now we turn our attention
to the section σ : C˜ \ S → X. In our setting σ can be written as
σ ( P ) = ( x ( P ), y ( P ), P ) ∈ π −1 ( P ) ∼= P2 \ DP .
Now it is a general fact that such a morphism has degreebounded by the height of its components: indeed the degreeof σ is the number of intersection points with a generic hyper˜ is embedded and thisplane in a projective space where σ(C)number is bounded by the sum of the heights of the components x and y. This proves thatdeg σ(C) ≤ HC˜ (u1 ) + HC˜ (y),68

4. The non-split casewhere u1 := x. The fact that the image σ( P) avoids the line
D2 means that the function u1 := x ∈ OS∗ , i.e. it is a unit,
and y ∈ OS , i.e. it is a regular function on the affine curve
C . Moreover, the condition that the image of σ avoids also
the conic D1 in every fiber means that we can define anotherS-unit u2 where
u2 = y2 − u21 − λu1 − 1.
Hence the units u1 , u2 and the S-integer y verify equation (4.5).

We will now work with equation (4.5) in order to describe its
solutions. Our goal is to prove the following
Theorem 4.4.2. With the notation above, every solution(y, u1 , u2 ) ∈ OS × (OS∗ )2of equation (4.5) satisfies one of the following conditions:(i) a sub-sum on the right term of (4.5) vanishes;(ii) u1 , u2 verify a multiplicative dependence relation of the form
u1r · u2s = µ, where µ ∈ κ ∗ is a scalar and r, s, are integers,
non both zeros such that max{r, s} ≤ 5;(iii) the following bound holds:˜ + 28H ˜ (λ) + 8H ˜ (λ).
max{ HC˜ (u1 ), HC˜ (u2 )} ≤ 212 58χS (C)CC
We will now follow the proof, given by Corvaja and Zannierof the constant case (Theorem 3.2.3 at page 50), deepening thedifferences between the present and the non-split situation.
The starting point is the description of a suitable notion ofderivatives of rational function over the curve C . This comesfrom Lemma 3.2.5 (page 52) where an appropriate differential
form is defined. We just notice that the Lemma refers to thecurve only without any reference to the bundle and hence canbe applied in all the cases under consideration.

4.4. Proof of the result

69

From now on the differential form ω and the finite set T appearing in Lemma 3.2.5 will be fixed and, for a rational func˜ we will denote by a0 the only rational function
tion a ∈ κ (C)such that d( a) = a0 · ω.
We consider now the derivative of a polynomial A ∈ κ [ X, Y ]calculated in a point u1 , u2 for some S-units u1 , u2 . One canprove that (see [CZ5] Lemma 3.7)

( A(u1 , u2 ))0 = B(u1 , u2 ),where
B( X, Y ) =

u10
u0
∂A
∂A
·X
( X, Y ) + 2 · Y( X, Y ).
u1
∂X
u2
∂Y

We will use this identity in order to deal with equation (4.5).
Lemma 4.4.3. Let
A( X, Y ) = X 2 + λX + Y + 1,
 0

u10 2
u1
u0
λ0
B( X, Y ) = 2
X +λ+
X + 2Y
u1
u1
λ
u2

(4.6)

˜ X, Y ], and let F ( X ) ∈ OS∪T [ X ], G (Y ) ∈be polynomials in OS∪T (C)[OS∪T [Y ] be the resultants of A( X, Y ), B( X, Y ) with respect to Y
and X, i.e. the polynomials

F (X) = X

2



u0
u0
2 1− 2
u1
u2




u10
u20
u0
λ0
−
+
λ− 2
+X
u1
u2
λ
u2


(4.7)

2
 0 2
u10
u0 u0
u1
u20
G (Y) = Y 2 −+Y(8 − λ2 ) + 1 2 (λ2 − 4)+
u1
u2
u1
u1 u2
 0



2
u10
u0
λ
+ λλ0
− 2
+
(4 − λ 2 ) + λ 02 .
(4.8)
λ
u2
u1
2



Then for every solution (y, u1 , u2 ) ∈ OS × (OS∗ )2 of (4.5) we have70

4. The non-split casey2 = A ( u1 , u2 ),2yy0 = B(u1 , u2 ).
Moreover the S-integer y divides both F (u1 ) and G (u2 ) in the ringOS∪ T .
Proof. Obviously equation (4.5) is exactly y2 = A(u1 , u2 ). Moreover A(u1 , u2 )0 = B(u1 , u2 ), so we have 2yy0 = B(u1 , u2 ) as
desired.
For the second fact we observe that, for the general theoryof resultants, F and G are linear combinations of A and B
with coefficients that are polynomials in OS∪T , concluding theproof.
Our next step will be to factor the polynomials F ( X ), G (Y ) in˜ ; this extension will bea suitable finite field extension of κ (C)a function field κ (D̃) for a cover D̃ → C˜. Besides, we will
estimate the Euler characteristic of the curve D̃ . From now on
we will suppose that the leading and the constant term of thepolynomial F ( X ), G (Y ) are both non zero.
Lemma 4.4.4. Given F, G, C˜, S, T as before, there exists a coverD̃ → C˜, of degree less or equal to four, such that the Euler characteristic of D̃ \ U verifies
˜ + 28H ˜ (λ) + 5 · max {0, 2g(C)˜ − 2}, (4.9)
χU (D̃) ≤ 53χS (C)C
where U is the set formed by the pre-images of the zeros of the leading and constant coefficients of F and G and the pre-images of S andT.
Proof. Our goal was to factor F ( X ) and G ( X ), so we define
the cover p : D̃ → C˜ by the property that κ (D̃) is the split˜ . From this definition itting field of F ( X ) · G ( X ) over κ (C)is straightforward that deg p is at most four, because κ (D̃) is4.4. Proof of the result71

˜ by the square roots of the discrimigenerated over p∗ (κ (C))
nants of the two polynomials (recall that F ( X ) and G ( X ) bothhave degree 2).
We will now bound the Euler characteristic of D̃ \ U via the
Riemann-Hurwitz formula; for this goal we need an estimateof the ramification points of the cover p. First of all we noticethat the ramification points are all contained in the zeros andpoles of the discriminants; moreover at any point the ramification index is at most two. The poles are contained in S ∪ T
and the number of zeros of the discriminants is bounded bytheir heights. The discriminant of F ( X ) is
Discr ( F ( X )) =

2

u0
u0
8 1 − 2 1 λ2 +
u1
u1
  0
2
u
− 2λλ0 + λ 1 + λ02 ,(4.10)
u1
u20
u2



2

( λ − 4) +

u20
u2



so its height (which can be estimated counting its possible
poles) is bounded by
 0
 0
u
u2
HC˜ ( Discr ( F ( X ))) ≤ 2HC˜
+ 2HC˜ 1 + 2HC˜ (λ0 ) + 2HC˜ (λ)
u2u1
˜ + 4H ˜ (λ).
≤ 6χS (C)C

Analogously we can look at the discriminant of G ( X )
2
u10
u0
λ2 (4 − λ2 ) + 1 λλ0 (8 − 2λ2 )+Discr ( G ( X )) =
u1
u1



3
u10
u0
+ λ 02 ( λ 2 − 4 ) + 2 2
λ2 (4 − λ2 )+u2u1
 0 2

u1
u10 020 2
20+λλ (λ − 8) + λ (4 + λ ) − λλ +u1u1
 0 4
 0 2u1u+
λ 2 − 2 1 λ 2 λ 02 + λ 04
(4.11)
u1
u1


u20
u2

2 

72

4. The non-split caseand bound its height in the same way, obtaining thatHC˜ ( Discr ( G ( X ))) is bounded above by


2HC

u20
u2





+ 4HC˜u10
u1

+ 4HC˜ (λ0 ) + 4HC˜ (λ) ≤˜ + 8H ˜ (λ).
≤ 10χS (C)C
Therefore the number of ramification points is at most
˜ + 12H ˜ (λ).
](S ∪ T ) + 16χS (C)C
We can now apply the Riemann-Hurwitz formula˜ − 2) + ∑ ( e P − 1).
2g(D̃) − 2 = (deg p)(2g(C)(4.12)P∈D̃

Here e( P) is the ramification index of p at P and thus (e P − 1)
is either zero or one. Now we can apply the above estimate of
the ramification points of p and we obtain that˜ + 12H ˜ (λ).
∑ (eP − 1) ≤ ](S ∪ T ) + 16χS (C)C(4.13)

P∈D̃

Consider now the set U ⊂ D̃ introduced in the statement of
the Lemma. We have that˜ · ] p (U ) .
]U ≤ [κ (D̃) : p? (κ (C))]
From this inequality and from (4.12) and (4.13) the quantity
2g(D̃) − 2 + ]U is bounded above by4.4. Proof of the result73
˜ − 2 + ] p(U ) + ](S ∪ T ) + 16χS (C)˜ + 12H ˜ (λ)(deg p) 2g(C)C˜ − 2 + ](S ∪ T ) + ]( p(U ) \ (S ∪ T ) += (deg p) 2g(C)˜ + 12H ˜ (λ)+ ](S ∪ T ) + 16χS (C)C˜˜ + 12H ˜ (λ).
≤ 4χS∪T (C) + 4]( p(U ) \ (S ∪ T )) + ](S ∪ T ) + 16χS (C)C

We have to bound the number ]( p(U ) \ (S ∪ T )), but the points
in the image of U that are not in S ∪ T are precisely the zerosof the leading and constant terms in F ( X ) and G ( X ). Again
we can estimate their number by looking at the height of theseterms. We obtain that


 0
u10
u
u20
]( p(U ) \ (S ∪ T )) ≤ HC˜ 2 −+ HC˜ 2 +
u1
u2
u2
 0




2
u1
u10
u20
2
02
+ 2HC˜ 2 −+ HC˜(4 − λ ) + λ
u1
u2
u1
 0 2u1
2
02
˜≤ 4χS (C) + HC˜(4 − λ ) + λu1
˜ + 4H ˜ (λ).
≤ 8χS (C)C

Taking this into account we can return to the previous inequality to finish our proof:

˜ + 32χS (C)˜ + 16H ˜ (λ) + ]S + ] T +χU (D̃) ≤ 4χS∪T (C)C˜+ 16χS (C) + 12H ˜ (λ)
C

˜ + 28H ˜ (λ) + 5] T + ]S≤ 52χS (C)C
˜ + 28H ˜ (λ) + 5 max{0, 2g(C)˜ − 2}.
≤ 53χS (C)C

74

4. The non-split case

The next step in the proof of our main result is an applicationof a theorem by Corvaja and Zannier concerning the “greatest
common divisor” of two rational functions on C˜ of the forma − 1 and b − 1 where a and b are units with respect to somespecified finite set (in our case the set will be U). In particular we are going to apply Corollary 3.2.9 from page 54 for asuitable choice of units a and b: these units will be chosen insuch a way that their heights will be “close” to the heights of
u1 , u2 and such that the sum appearing in the statement of theprevious Theorem gives an upper bound for ∑v∈D̃\U v(y). We
begin by proving the following
Lemma 4.4.5. Let (u1 , u2 , y) be a solution of equation (4.5)(recall that we are supposing that the leading and constant coefficients of F, G are both non zero). Let D̃ , U as before. Then there
exist U-units a, b ∈ κ (D̃) such that the quantity|max{ HD̃ ( a), HD̃ (b)} − max{ HD̃ (u1 ), HD̃ (u2 )}|(4.14)

˜ + 8H ˜ (λ) andis bounded above by 32 · χS (C)C
1∑ min{v(1 − a), v(1 − b)} ≥ 4 · ∑ v(y).

v∈D̃\U(4.15)

v∈D̃\U

Moreover, a = u1 α−1 , b = u2 β−1 for suitable α, β such that
F (α) = G ( β) = 0.
Proof. Being the field κ (D̃) defined as the splitting field for thepolynomial F ( X ) · G ( X ) we can write the two polynomials as

 0
u1
u20
( X − α) · ( X − ᾱ),F(X ) = 2 −
u1
u2
 0
2
u1
u20
G(X ) = 2 −( X − β) · ( X − β̄).
u1
u2
We claim that the roots α, ᾱ (resp. β, β̄) of F (resp. G) are
U-units. This follows from the definition of U (see Lemma

4.4. Proof of the result75(4.4.4)), because the leading and constant coefficients of thetwo polynomials are U-units. We consider now the followingu0u0 polynomials obtained from F and G dividing by αᾱ 2 u11 − u22u0
u 0 2and β β̄ 2 u11 − u22 respectively, i.e. the polynomials
F ( X ) := ( Xα−1 − 1)( X ᾱ−1 − 1)
G ( X ) := ( Xβ−1 − 1)( X β̄−1 − 1).
Now, by Lemma (4.4.3), the U-integer y divides both F (u1 )and G (u2 ), and hence it divides the polynomials F (u1 ) andG (u2 ) in the ring of U-integers. From this it follows that∑ min{v(u1 α−1 − 1) + v(u1 ᾱ−1 − 1), v(u2 β̄−1 − 1) + v(u2 β̄−1 − 1)}v∈D̃\Uis bounded below by∑ v ( y ).

v∈D̃\UWe want to analyze the left side term of the last inequality: observe that for every fourple of rational functions W1 , W2 , Z1 , Z2
one has (we omit the valuations)∑ min{W1 + W2 , Z1 + Z2 } ≤ ∑ min{W1 , Z1 } + ∑ min{W1 , Z2 }+vvv+ ∑ min{W2 , Z1 } + ∑ min{W2 , Z2 } ≤ 4 ∑ min{W̃, Z̃ },vvv

for suitable W̃ ∈ {W1 , W2 } and Z̃ ∈ { Z1 , Z2 }. In our case we
obtain that there exist U-units a ∈ {u1 α−1 , u1 ᾱ−1 } and b ∈{u2 β−1 , u2 β̄−1 } such that:4∑ min{v(a − 1), v(b − 1)} ≥ ∑ v(y),v∈D̃\Uv∈D̃\Uproving (4.15). Next we want to prove that the heights ofthese U-units a, b are “close” to the heights of u1 , u2 . We observe that the difference appearing in the left side term of

76

4. The non-split case

(4.14) is bounded by the maximum of the D̃ -heights of theroots of F and G. Again we bound these heights by estimating their possible poles. It is then sufficient to observe that thepoles of the roots α, ᾱ (resp. β, β̄) are either zeros of the leading coefficient or poles of the constant term of the polynomial
F (resp. G). Hence


 0
u10
u20
u
max{ HD̃ (α), HD̃ (ᾱ)} ≤ HD̃ 2 −+ HD̃ 2
u1
u2
u2

 0
 0
0
u
u
u
≤ 4HC˜ 2 1 − 2 + 4HC˜ 2
u1
u2
u2
˜ .
≤ 8χs (C)In the same way we get the quantity max{ HD̃ ( β), HD̃ ( β̄)} isbounded above by
2
 0 2

u10
u1
u20
2
02
HD̃ 2 −+ HD̃(4 − λ ) + λ
u1
u2
u1
 0
 0 2
2

u
u1
u0
≤ 4HC˜ 2 1 − 2 + 4HC˜
(4 − λ 2 ) + λ 02
u1
u2
u1
˜ + 8H ˜ (λ).
≤ 32χs (C)C

In order to apply Theorem (3.2.9) we need an upper boundfor ∑v∈D̃\U v(y) in terms of the heights of u1 , u2 . This boundis obtained by an application of a theorem by U. Zannier in[Zan] which reads as follows:
Theorem 4.4.6 (Zannier). Let D̃ , U as before, m ≥ 2 an integer, θ1 , . . . , θm U-units such that no subsum of θ1 + · · · + θmvanishes. Then the U-integer θ1 + · · · + θm satisfies
 
m
v
(
θ
+
·
·
·
+
θ
)
≥
H
(
θ
:
·
·
·
:
θ
)
−
χU (D̃).
m
m
1
∑
D̃ 1
2
v∈D̃\U4.4. Proof of the result

77

We are going to apply this Theorem to the U-integery = u21 + λu1 + u2 + 1,
using the fact thatHD̃ (u21 : λu1 : u2 : 1) ≥ max{2HD̃ (u1 ), HD̃ (u1 ) + HD̃ (λ), HD̃ (u2 )}≥ max{ HD̃ (u1 ), HD̃ (u2 )}.
In particular, assuming that no subsum of the right term ofequation (4.5) vanishes, we obtained the following
Lemma 4.4.7. For every solution (y, u1 , u2 ) of (4.5) such thatno subsum of the right term vanishes, one hasHD̃ (y) ≥∑ v(y) ≥ max{ HD̃ (u1 ), HD̃ (u2 )} − 6χU (D̃).

v∈D̃\U

Now we put together this last inequality with the results ofLemma (4.14) and we obtain that, for every solution of (4.5)
there exist U-units a, b such that the sum∑ min{v(a − 1), v(b − 1)}v∈D̃\U

is greater or equal than
1
4




˜
max{ HD̃ ( a), HD̃ (b)} − 6χU (D̃) − 32χS (C) − 8HC˜ (λ) .

˜ ≤ χU (D̃) we obtain that the same
Using the fact that χS (C)quantity is bounded below by

1
4




˜
max{ HD̃ ( a), HD̃ (b)} − 38χS (C) − 8HC˜ (λ) .

(4.16)

We can now apply Theorem (3.2.9) to deduce the following

78

4. The non-split case

Proposition 4.4.8. Let (y, u1 , u2 ) ∈ OS × (OS∗ )2 be a solution of equation (4.5) such that no subsum of the right term vanishes, and the leading and constant term of the polynomials F, G are∗ as
not zero. Let D̃ , U be as defined in Lemma (4.4.4) and a, b ∈ OU
defined in Lemma (4.4.5). Then either
˜ + 28H ˜ (λ) + 8H ˜ (λ)
max{ HC˜ (u1 ), HC˜ (u2 )} ≤ 212 58χS (C)C
C(4.17)or a, b verify a multiplicative dependence relation of the formar · b s = 1
for integers (r, s) ∈ Z2 \ {0} with
max{|r |, |s|} ≤ 5.

(4.18)

Proof. We suppose that inequality (4.17) does not hold andwe want to prove the dependence relation for a, b. In order toapply Corvaja and Zannier Theorem (3.2.9) we are going toshow that the left-hand side of (4.17) is greater than the righthand side of (3.3). Our starting point is˜max{ HC˜ (u1 ), HC˜ (u2 )} > 2 · 58 · χS (C) + 28HC˜ (λ) + 8HC˜ (λ).
12From Lemma (4.4.4) we know that˜ + 28H ˜ (λ)χU (D̃) ≤ 58 · χS (C)C
and so we obtain thatmax{ HC˜ (u1 ), HC˜ (u2 )} > 212 χU (D̃) + 8HC˜ (λ).
Remember that our aim is to apply Theorem (3.2.9) and so
we need to work with the maximum of the heights of a, b.

4.4. Proof of the result

79

For this reason we apply (4.14) which estimates the closeness of H (ui ) and H ( a), H (b) and, using HC˜ ≤ HD̃ we getmax{ HD̃ ( a), HD̃ (b)} is bounded above by
˜ − 8H ˜ (λ)
max{ HD̃ (u1 ), HD̃ (u2 )} − 32χS (C)C≥ max{ HD̃ (u1 ), HD̃ (u2 )} − 32χU (D̃) − 8HC˜ (λ).
From these last two inequalities we obtain the lower boundmax{ HD̃ ( a), HD̃ (b)} ≥ (212 − 32)χU (D̃).

(4.19)In order to simplify the notation we put H = max{ HD̃ ( a), HD̃ (b)}and χ = χU (D̃). We claim that∑ min{v(a − 1), v(b − 1)} > 3 · 2 H χ
1
3

2
3

1
3

(4.20)

v∈D̃\UTo prove the claim we observe that, from (4.16), it is enoughto show that12
1
1
H − 38χ > 3 · 2 3 H 3 χ 3 .
4
We define the function
f (t) =

1 2
1
1
t − 3 · 2 3 t 3 χ 3 − 38χ
4and we notice that our claim is equivalent to f ( H ) > 0. Now
the function f is an increasing function for t ≥ 210 χ, therefore
it is enough to prove it for H = (212 − 32)χ > 210 χ. Hencethe claim is equivalent to12
1 12(2 − 32)χ − 3 · 2 3 (212 − 32) 3 χ > 38χ
4With some algebraic manipulations one gets80

4. The non-split case

1
2
1 12
(2 − 32)χ − 3 · 2 3 (212 − 32) 3 χ =
4


10
2 1 5
1
1
7
7
3
3
3
3
3
2 ((2 − 1) − 3 · 2 χ =2 (2 − 1)
4


13
1
− 13
6
3
40 · 6 · 2 · 2 − 3 · 2 χ =
1540 · 6 · 2 3 (2 3 − 3) χ >140 · 6 · 1 · χ
6which proves the claim. Now we can apply Theorem (3.2.9)
which implies that a, b verify a multiplicative dependence relation of the form ar bs = 1 for some integers r, s not both zero.
The same Theorem gives the bound (3.4) and hence, together
with (4.19) and (4.16), we obtainH
1
1> H − 10χ > H
max{|r |, |s|}4
5
Therefore we get max{|r |, |s|} ≤ 5, as desired.
The conclusion of Proposition 4.4.8 gives us a multiplicative
relation of dependence between a, b instead of u1 , u2 . However this relation is guaranteed by Lemma 3.14 in [CZ5] which gives us the following result:Lemma 4.4.9 ([CZ5]). In the previous notation, if a multiplicative relation of the form ar · bs = µ holds for a constant µ ∈ κ,
then either one between a and b is constant or u1 , u2 satisfy a multiplicative dependence relation of the same type.
Now we go back to Theorem (4.4.2): here we should take careof the constant term of the polynomial G in a different way
as in the constant case. In detail the vanishing of this term
does not directly imply an explicit bound for the degree of
the images f (C) as in the split function field case; here we4.4. Proof of the result

81should apply again the whole machinery in order to explicitly
find the unit u1 and so reduce the problem to equation y2 =µ + u2 + 1, which was already solved in the split case andgives the desired bound. For readability reasons we split theproof of Theorem (4.4.2) in two cases: Lemma (4.4.10) for thecase in which the constant coefficient of G is not zero, andLemma (4.4.11) for the other case. Clearly the two lemmas
together gives Theorem (4.4.2).
Lemma 4.4.10. Suppose that the constant term of the polynomial G does not vanish, i.e., with the notation of 4.4.2, every solution (y, u1 , u2 ) ∈ OS × (OS∗ )2 of equation (4.5)
y2 = u21 + λu1 + u2 + 1
satisfies also


u10
u1

2

2

(4 − λ ) + λ

2



λ0
λ

2

6= 0.

(4.21)

Then one of the following conditions holds:(i) a sub-sum on the right term of (4.5) vanishes;(ii) u1 , u2 verify a multiplicative dependence relation of the form
u1r · u2s = µ, where µ ∈ κ is a scalar and r, s, are integers,
non both zeros such that max{r, s} ≤ 5;(iii) the following bound holds:˜ + 28H ˜ (λ) + 8H ˜ (λ).
max{ HC˜ (u1 ), HC˜ (u2 )} ≤ 212 · 58 · χS (C)C
C
Proof. We start assuming that (i), (ii) and (iii) are not satisfied and we are going to find a contradiction. First of all we
note that, if (i) is not satisfied, no subsum of (4.1) can vanish.
Moreover the polynomials F and G defined in 4.4.3 could not
be constant because the vanishing of their leading coefficientswould imply some multiplicative relation between u1 and u2
which is excluded by (ii). The same is true for the constant coefficient of F (which is u20 /u2 ): it cannot be zero otherwise u2

82

4. The non-split casewould be constant; moreover, by our assumptions, the same
holds for the constant coefficient of G. Hence both F and G
are non constant polynomials whose constant coefficients are
not zero.
Since we excluded the case where the leading and constantcoefficients of F and G vanish, we can apply (4.4.8) and obtaina multiplicative relation between a = u1 α−1 and b = u1 β−1 ;this follows from the fact that inequality (4.17) is excluded by(iii). From this relation, applying (4.4.9), we get that either aor b is constant or u1 and u2 verify a multiplicative relation of
the same type. The former case would imply that the heightof u1 (or u2 ) would be the same as the height of α (resp. β) so
˜ (resp. 32χS (C)˜ ); butit would be lesser or equal than 8χS (C)this contradicts our assumption that (iii) does not hold andhence it is excluded. The latter case is precisely (ii) that was
assumed to be false. In both cases we get a contradiction andthis concludes the proof.

Lemma 4.4.11. Suppose that the constant term of the polynomial G vanishes, i.e., with the notation of 4.4.2, every solution(y, u1 , u2 ) ∈ OS × (OS∗ )2 of equation (4.5) satisfies also


u10
u1

2

(4 − λ2 ) + λ2



λ0
λ

2

= 0.

(4.22)

Then one of the following conditions holds:(i) (y, u1 , u2 ) satisfy an equation whose solutions verify conclusions of Theorem (4.4.2).
(ii) u1 , u2 verify a multiplicative dependence relation of the form
u1r · u2s = µ, where µ ∈ κ is a scalar and r, s, are integers, notboth zero such that max{r, s} ≤ 5;
Proof. The first trivial case is the case in which λ is constantwhich is excluded since we are assuming that the threefold

4.4. Proof of the result

83

defined by λ is not trivial. The second case is the case in which
λ is a non constant S-unit. In this case we obtain, in the ring
OS , the following identity (here we recall that can enlarge S
so that it contains every point for which λ = 2):


u10
u1

2

λ2
=−
4 − λ2



λ0
λ

2
.

(4.23)

Now we observe that the ring OS∗ is finitely generated modulo
constants, so every u1 ∈ OS∗ is of the form µ · v1a1 · · · vhah forsome µ ∈ κ and v1 , . . . , vh ∈ OS∗ . Therefore we havehv0u10
= ∑ ai i .
u1
vi
i =1

Being λ ∈ OS∗ the right-hand side of equation (4.23) could
also be expressed in the vi and their derivatives; in particular
4.23 becomes an equation in the unknown ai and this equation will have a unique (for given λ and S) solution in the ai .
Hence u1 will be uniquely determined up to a constant factor
and therefore its height will be a constant. So we can assume
that u1 = a f for a constant a ∈ κ and a fixed S-unit f . This
leads to consider equation y2 = a2 f 2 + λa f + u2 + 1. We claimthat this case gives (i). The claim follows from a repetition of
all the considerations done until now for equation (4.5): we
obtain the same estimates with different polynomials F̃, G̃.
Again we look at the vanishing of the constant and leading
coefficients and this time we found that the case in which theconstant coefficient of the new polynomial G̃ vanishes gives
us either u1 = 0 or u1 = f where a = 1. In both cases thisreduces the problem to the equation y2 = µ + u2 + 1, where µ
is now fixed, which has already been treated in the split function field case and gives (i). The case in which the constantterm of G̃ is not zero is precisely one of the cases of (i) andthis concludes the proof of the claim.

84

4. The non-split case

Finally we prove Theorem (4.1.1) using the previous Theorem.
Proof of Theorem (4.1.1). As in Lemma (4.4.1) a section
σ : C˜ \ S → X
will be of the form
σ : P 7 → ( u1 ( P ), y ( P ), P )where the S-unit u1 and the S-integer y verify equation (4.5)
for a S-unit u2 . In this setting we can apply Theorem (4.4.2)
and conclude that one of (i),(ii),(iii) holds. Let us analyze every case.
• In the first case (i) we have that some sub-sum of u21 +λ( P)u1 + u2 + 1 will vanish. Hence σ(C) is either a line
or a conic and its degree verifies the bound (recall thatbeing λ non constant its height is at least one).
• In the second case (ii) we have a multiplicative relation between the two S-units of the form u1r = u2s · µ
for a scalar µ ∈ κ and two integers r, s with absolutevalue lesser or equal than 5. From this it follows thatdeg σ(C) ≤ HC˜ (u1 ) + HC˜ (y) ≤ 20 and again the boundis verified.
• In the last case (iii) we have max{ HC˜ (u1 ), H
 C˜ (u2 )} is12
˜
bounded above by 2 · 58 · χS (C) + 28HC˜ (λ) + 8HC˜ (λ)
from which we obtain the desired bound.

III
The split case5
Approaches to fewer
components
Up to this point we have discussed cases of Lang-Vojta Conjecture in which the number of components of the divisor at
infinity D were at least three. The aim of the last part of this
Thesis is to develop a strategy to deal with complements ofdivisor with fewer components, or even irreducible. We begin by giving the following definition:
Definition 5.0.12. Given a normal crossing divisor D in asmooth projective variety X we call logarithmic irregularity of
the couple ( X, D ) the number of irreducible components minus 1.
Equivalent formulation, similar to the compact case, can begiven by defining logarithmic irregularity as the dimensionof the space of sections of the logarithmic cotangent bundle.
Previous Theorems like Theorem 3.1.1 and Theorem 3.2.3 andmore generally Theorem 3.3.2 imply that, for P2 , the split caseof Lang-Vojta Conjecture for function fields is known when
the logarithmic irregularity of P2 and the divisor D is greateror equal to 2, i.e. is greater or equal to the dimension. The firstquestion we want to address is whether this methods can beapplied to study the case of logarithmic irregularity strictly
smaller than the dimension or not.

88

5.1

5. Approaches to fewer components

Limits of Corvaja and Zannier methods

The methods applied in chapter 3 and chapter 4, coming fromideas of Corvaja and Zannier, extends to Theorem 3.3.2 atpage 56 (and in particular for the projective plane Theorem
3.3.3 at page 56). However it is clear that the key point of allthese strategies is, explicitly or implicitly, the existence of afinite map to G2m . Such existence follows from the presence ofat least three components of the divisor at infinity. Actually
the equations of the three components of the divisor D defines
such a map as sketched in Remark 3.3.1 at page 56. Once theexistence of the map is proved, one constructs an “extra component” coming form the ramification and then reduces thealgebraic hyperbolicity to a generalized S-units equation insuch a way that the conclusion follows from an application ofBrownawell-Masser Theorem (see Subsection 3.3.1 at page 57
for more details).
The question we are interested in now is whether this strategy
could be modified to deal with the case of logarithmic irregularity strictly lesser than 2. As discussed above this seems
unlikely. Let us focus on a specific key example in which thisdifficulty is showed:
Example 5.1.1. Let D be a divisor in P2 consisting of two conics in general position. Let f 1 , f 2 be the equations of the two
conics. It is defined a natural map from the complement P2 \D to Gm by
P 7→

f 1 ( P).
f 2 ( P)

One can also prove that Gm is the generalized Albanese Variety of P2 \ D which in particular implies that every map
P2 \ D to the two dimensional torus factors through the pre-

5.1. Limits of Corvaja and Zannier methods

89

vious map to Gm , i.e.
P2 \ D

GG
GG
GG
GG
G#

Gm

/ G2
= m
|
|
|
|
||
||

Thus there is no natural map from P2 \ D → G2m allowing toapply Theorem 3.3.3. At the same time once can see how the“algebraic” approach developed in Chapter 4 and Chapter 5
can be applied neither. In facts one could try to mimic theconstruction of the corresponding equations in S-units and Sintegers, but in this case, having only the equations of the two
conics defining the complement, one can build up just one Sunit and two S-integers. The corresponding equations cannot
be handled as the one considered in Chapter 5. As an example
of a situation that can occur one could consider the following
equationx2 = P(u, y),in which x, y are S-integers and u is an S-unit for a finite setof points S in a smooth curve C˜ and P is a polynomial. Differentiating with respect to suitable differential form defining
on the curve one obtain the following equation
2xx 0 =u0
∂P(u, y)∂P(u, y)
·u
+ y0
,
u
∂u
∂y

where now the term multiplying the derivative with respect
to y has height that cannot be bounded independently of y,because y is not a S-unit.
Previous example shows how the case of logarithmic irregularity strictly less than 2 are cannot be recovered from the
known cases of bigger irregularity. In particular, the differentiation argument used in the proof of Theorem 3.2.3 andTheorem 4.1.1, that led to an application of a gcd theorem of
S-units, does not seems to be recovered here. Namely the lackof a third irreducible component of the divisor D translate

90

5. Approaches to fewer components

into the absence of a natural defined S-unit. This led to consider more difficult Diophantine equations for which, at thepresent, no general methods can be applied to describe theirsolutions. At the same time, the same problem arises when
one tries to construct a map to the two dimensional algebraic
torus, in order to apply Theorem 3.3.2: the equation of a third
irreducible component appears to be fundamental for such a
map to exists and hence we cannot see any way to recover results for quartics with less than three irreducible components
using the constructions and results obtained by Corvaja andZannier.
We end this section by noticing that such limitation appears
also when considering complements of plane curves withhigher degree. In fact, Theorem 3.3.3 applies to complementsof curves of arbitrary degree, provided that the curve has atleast three components and the degree is greater than four.
For this reason it is natural to consider other approach for
dealing with affine surface with log irregularity lesser than
two.

5.2

Ideas from deformations and curve counting

A first systematic treatment of algebraic hyperbolicity for complements of normal crossing divisor in P2 where the log irregularity is strictly smaller than 2 can be found in Xi Chen’s article [Che3] (and previously for surface in the pre-print [Che2]).
Although with a different goal in mind, i.e. Kobayashi conjecture on hyperbolicity of complements of nc divisors in P2 , theideas presented in the papers have some similarities with theapproach taken in this Thesis. Therefore we will devote thissection to an introduction to Chen’s ideas and to its limitation
to an extension to degree four divisors.

5.2. Ideas from deformations and curve counting

5.2.1

91

Chen’s argument

The main idea of [Che3], already present in [Che2] is the following: algebraic hyperbolicity for complements P2 \ D ismuch easier to prove when D is reducible. Then for a generic
D the natural strategy would be to degenerate it to a “highly”
reducible divisor, i.e. a union of lines in general position andthen try to recover the same result using the knowledge of thecase of the union of lines. It turns out that this strategy gives
the desired result also in higher dimension, however we are
not going to focus on this important aspect of Chen’s work
and we focus only on the two dimensional case.
In order to state precisely the results let us fix some notation first: in this section S will be a normal projective variety with canonical singularities (although in our application
S will always be P2 hence non-singular), N1 (S) will denotethe free abelian group generated by 1-cycles modulo numerical equivalence, and N 1 (S) will denote the group of divisor
modulo the same equivalence. We call a function φ : N1 → R
an additive function on N1 (S) if φ ∈ Hom( N1 (S), R). Since inthe case S non singular one has Hom( N1 (S), R) = N 1 (S) ⊗ R,
an additive function φ corresponds to an R-divisor D andφ(C) = C · D.
The first result proved by Chen is a generalization of Theorem
3.1.1 to arbitrary surface and sufficiently reducible divisor.
Theorem 5.2.1 (X. Chen). Let S be as above, B = ∑in=1 Bi an
effective divisor with normal crossing singularities, F a curve on S.
Assume that Bi is a very general member of a base point free linear
system PLi for each i ≥ 1, while B0 and F are fixed curves, meeting
properly in smooth points of B0 . Let φ be a function
φ : Z1 := {1 − cycles on S} → R,
for which there exists e ∈ R such that
1. (KS + B − Bi ) · C ≥ eφ(C) for all i ≥ 1 and all C ⊂ S
non-rigid;

92

5. Approaches to fewer components

2. 2g(C) − 2 + ( B − B0 ) · C ≥ eφ(C) for all C reduced;
3. χ T (C) ≥ eφ(C) for all irreducible components of F, whereT = ν−1 (C ∩ B) and ν is the normalization of C .
Then for all reduced irreducible curves C in S, not contained in B
the following holds:
χ T (C) ≥ eφ(C),where T is as above.
We note that Chen does not use the notion of Euler Characteristic for affine curves as used throughout this Thesis, butrather defines explicitly an intersection multiplicity, that hedenotes by iS (C , D ). We used previous notation in order to beconsistent with the rest of the Thesis.
Theorem 5.2.1 gives the already known result of algebraic hyperbolicity for the complement of at least 5 lines in P2 : is it
sufficient to apply the Theorem with B0 = F = ∅, B − B0 =L1 , . . . , Ld with d ≥ 5 and φ = deg. We stress however thatthe requirement d ≥ 5 is crucial since:
KP2 + B − Bi ∼ (d − 4) H,
with H a line in P2 . This feature with be important in the sequel: in particular it gives first evidence that the argument weare going to describe, using Theorem 5.2.1, cannot be applied
directly to complements of quartics.
The key idea of Chen’s construction is to take Theorem 5.2.1
as a starting point and reducing the proof of algebraic hyperbolicity for complements of irreducible divisor to the Theorem via degeneration. This idea is made explicit in the following
Theorem 5.2.2. In the notation above, let{ I1 , I2 , . . . , Im } be apartition of {1, 2, . . . , n}, D0 = B0 and let Dk be a very general
member of P(⊗i∈ Ik Li ) for every k > 1. Let D be D0 ∪ D1 ∪ · · · ∪Dm . If φ : N1 (S) → R is an additive function such that there
exists a real number e such that

5.2. Ideas from deformations and curve counting

93

1. (KS + B − Bi ) · C ≥ eφ(C) for all i ≥ 1 and all C ⊂ S
non-rigid;
2. 2g(C) − 2 + ( B − B0 ) · C ≥ eφ(C) for all C reduced;
3. χ T (C) ≥ eφ(C) for each irreducible component C ⊂ F, withT = ν−1 (C ∩ D );then for all irreducible curves C in S not contained in D the following holds:
χ T (C) ≥ eφ(C).
The Theorem formalized the idea of deformations of the divisor D: here the components Dk are degenerated to the union∪ Bj for j ∈ Ik in such a way that Theorem 5.2.1 can be applied.
Choosing as before S = P2 , B0 = F = ∅, Bi a line in P2 foreach i ≥ 1, D a very general curve of degree d in P( H 0 (OS ))and φ = deg one get the following:
Corollary 5.2.3. Given a very general plane curve D of degreed ≥ 5, for all reduced irreducible curves C ⊂ P2 not contained inD, denoted by T = ν−1 (C · D ), with ν the normalization of C , thefollowing holds:
χ T (C) = 2g(C) − 2 + #T ≥ (d − 4) deg C .
In particular for all very general plane curves D of degree d ≥ 5,
P2 \ D is algebraic hyperbolic.
We note again that the bound degD ≥ 5 is strict since bothin Theorem 5.2.2 and in Corollary 5.2.3 the statement and theproof will be false if we consider D to be a generic quartic.
Moreover, in Chen’s paper, his definition of algebraic hyperbolicity refers to what we called strong algebraic hyperbolicity (see page 41); this in particular implies that Theorem
5.2.2 gives that no map Gm → P2 \ D exists provided thatdeg D ≥ 5 and D is a very general member of OP2 (d).
Idea of the Proof of Theorem 5.2.2 As pointed out above
the main idea of the proof is to degenerate D to a union ∪ Bi

94

5. Approaches to fewer components

for which Theorem 5.2.1 can be applied. What actually isdone is to degenerate one component of D at the time, while
keeping the remaining fixed, and then applying an induction
argument on the number of irreducible components.
Let X = S × ∆, with ∆ the unit disk parametrized by t, andlet W be an effective divisor in the family X such that
• W is the sum of m components W (1) , . . . , W (k) ;• for each k 6= 1, W (k) restricted to each fiber is Dk ;• W (1) is a pencil of curves in P(⊗i∈ I1 Li ) where the cen(1)tral fiber W0 = B1 ∪ D10 is reducible, while the generalfiber is a general element of the linear series.
Basically the family consider the case in which D2 , . . . , Dm are
fixed D1 moves in a family W (1) in which the general fiber
Wt1 is a general member of the linear series P(⊗ I1 L), and thespecial fiber over t = 0 reduces to a union B1 ∪ D10 . In thissetting one can consider a family of curves Y → ∆, flat over∆, with a proper map π : Y → X and a commutative diagram
Y

π

/X

α


/∆



∆

with α a base change of order α. Up to applying semistablereduction one can assume that:• Y is smooth and irreducible and Y0 is nodal;• π |Yt maps Yt birationally onto its image for all t 6= 0;• π (Y ) meets W properly;• Yt ∩ π −1 (W ) extends to disjoint sections of Y → ∆;• Y → X is minimal with the above properties.
The conclusion of the Theorem is equivalent to prove:χ T (Yt ) ≥ eφ(π (Yt ))5.2. Ideas from deformations and curve counting95

for t 6= 0 assuming that
χ T0 (C) ≥ eφ(C),with C ⊂ S irreducible not contained in W0 and where T is
the set of inverse images in the normalization of Yt of the intersection with Wt and T0 are the inverse image, in the normalization of C , of the intersection with W0 . Now one candefine, for each component Γ of Y0 , a finite set of points σ(Γ)
which counts the point of intersection in Γ coming fromlim Yt ∩ π −1 (W )t →0and the nodes of Y in Γ. Then the following holds:
χ T (Yt ) = ∑(2p a (Γ) − 2 + #σ(Γ)).
Γ

Then one is reduced to show that
2p a (Γ) − 2 + #σ(Γ) ≥ eφ(π (Γ)).
At this point Chen studies separately the case in which Γ iscontracted, π (Γ) intersect properly W0 and finally, the mostdelicate case, when the support of π (Γ) is contained in W0 .
In both cases one can recover the desired bound for the Euler
Characteristic determining the set σ(Γ) via a subtle blowup
construction and a lower bound for the ramification of the
reduction map of the normalization of each component Γ.

5.2.2

Limits of Chen methods

We end this Chapter by a brief analysis of the features ofChen’s result and proof that will resemble part of ideas developed in the next chapter.
First of all the most difficult step in the above proof is to deal
with curves, or components of curves, that are mapped to acomponent of the divisor W0 . In facts, as described in Chen’s96

5. Approaches to fewer components

paper, one can look at the family π : Y → X as a family of
pre-stable maps with marked points given by the intersection
with the divisor W. With this picture in mind one should deal
with the possible degeneration that the image of such familyof maps could acquire in the fiber over t = 0. This is the hardpart in which a careful description and control of the set σ(Γ)
plays a fundamental role. The use of pre-stable map however
suggests that some moduli space of pre-stable maps could enter the picture allowing a more abstract reformulation of the
problem of hyperbolicity. This will be the path we are going
to follow in the last part of this Thesis.
Secondly we note how the deformation argument sketchedabove is really natural when dealing with hyperbolicity questions; Namely, in many cases (if not all the known cases), a
result for complements of reducible divisor, usually union of
hyperplanes in generic position, is already available, or canbe recovered in an easier way. Using this knowledge one
could try to obtain a result for a generic irreducible divisor
degenerating it to a union of sufficiently many components
having normal crossing and recover the desired conclusionfrom the known results in the reducible case. However thisstrategy works only for algebraic hyperbolicity and cannot be
extended to Kobayashi hyperbolicity, which was the starting
point of Chen’s analysis: unfortunately Kobayashi Hyperbolicity is not an algebraic condition on varieties.
Thirdly we want to stress the fact that the difficulty in dealing with the set σ as well as with the set S used in previous chapter, is that its cardinality depends only on the number of points of intersection but not on the multiplicities of intersection. Moreover, for non ordinary multiple points, themultiplicity coming from the singularity is not detected either. Hence one can say that the major difficulty in the previous construction is that, when dealing with the family of
pre-stable maps Y → X, there is no control on the multiplicities of intersection in the family.

5.2. Ideas from deformations and curve counting

97

Finally, from the fact that Chen was interested in strong algebraic hyperbolicity and not on log-general type surface, partof its method cannot simply be extended to deal with complements of very generic quartics in P2 . In particular his stronger result for complements of very generic curves of degree
at least 5, together with its generalization to higher dimension, provides a strategy to prove that there are no Gm immersion on the varieties considered, a result that we know to befalse in a generic log-general type surface (see Remark 2.2.7
at page 42). Moreover, all the results in its paper requires avery generic hypothesis that makes such results weaker thanthe corresponding one obtained by Corvaja and Zannier inthe three component case.
From these observation the idea of the present work is to takea different approach to this problem via Logarithmic Geometry. This will allow to control in an efficient way the multiplicities of intersection, incorporating them as a part of the data ofthe deformation. Namely Logarithmic Geometry will give us
tools that consent to fix such multiplicities in a family of stablemaps, once we consider the family as a family of logarithmic
maps. Then we will, in some sense, extend Chen’s deformation arguments for study algebraic hyperbolicity, in the weaksense, for complements of very generic quartic in P2 . The role
played by Theorem 5.2.1 will be taken by Corvaja and Zannier’s Theorem 3.2.3. Despite the naturality of a logarithmic
approach, we will need a heavy machinery coming form thetheory of Logarithmic Stable maps to Deligne-Falting pairs,
as developed by Qile Chen and Dan Abramovich. We will
devote all the next chapter to recall the basic facts we needfor state the mail result for complements of quartics.

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5. Approaches to fewer components

6
Logarithmic Geometry
Logarithmic Geometry can be traced back to ideas of Fontaine
and Illusie; the first systematic discussion of its foundations
has been given by Kazuya Kato in [Kat2]. The theory has been
applied with success both in Arithmetic Geometry (works by,among others, Faltings, Kato and Tsuji) and in Algebraic Geometry (related to, Toric Geometry, (p-adic) Hodge Theory,
Moduli Spaces and others). In this chapter we will recall thebasic definitions needed for fully describing the ideas of thenext chapter. For more detailed and extensive introduction tothis fascinating theory we refer the reader to Arthur Ogus lectures [Ogu], to the first sections of [ACG+ ] and to [Ill]. This
chapter is organized as follows: the first section will recall
the basic notion of logarithmic structure, logarithmic schemes
and maps moving from the pivotal example of a normal crossing divisor in a smooth variety. In the second section the focus
will be the notion of logarithmic curve which is the naturalgeneralization of pre-stable curve. We are going to briefly describe the notion of stable logarithmic curve giving evidence
that this notion naturally generalizes usual stable curves: inparticular we will show that log curves encodes naturally amarked structure as well as some constrains on the possible
singularities. Then we will see how it is possible to give e logarithmic structure to any pre-stable curve in a canonical way.

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6. Logarithmic Geometry

The last section will be dedicated to log stable maps, a generalization and an extension of usual stable maps originatedfrom the work of Kontsevich [Kon]. In this direction we willconcentrate on the moduli of stable maps and its analogous
in the logarithmic category, the stack of log stable maps in theparticular case in which the target log scheme is a Deligne
Falting pair, a structure to which belongs the couple we aremostly interested in, i.e. P2 and a reduced simple normal
crossing divisor D.
In the context of this Thesis the importance of Logarithmic
Geometry, and more in detail of log stable maps and theirmoduli, comes from the fact that they provide the appropriate framework in which Theorem 3.2.3 can be extended. Actually, the reformulation and extension of Corvaja and Zannier’s Theorem in terms of Logarithmic Geometry will be thekey point that will allow to extend the result to complements
of more general divisors.

6.1

Background and basic constructions

Throughout this Thesis we will assume that all monoids, i.e.
semi-groups with unit, are commutative and all morphisms
between monoids preserve the unit elements. The structure
sheaf of a scheme will be viewed as a sheaf of monoids undermultiplication, unless otherwise specified.
We begin by the motivating example of a normal crossing divisor in a smooth variety: this is not only one of the main historically motivation of Logarithmic Geometry, but it is also
the one that we will use the most in the following parts of this
Thesis.
Example 6.1.1. Let X be a smooth variety (or more generally aregular scheme) and let D be a normal crossing divisor on X.
If U denotes the complement X \ D, one can try to describe
the invertible elements of the ring O X (U ), i.e. regular func-

6.1. Background and basic constructions

101

tion of X that are invertible when restricted to U. In a morefunctorial way one can try to define a sheaf M on X that associates to each open subset V of X the set M(V ) of regularfunctions of X whose restriction to V \ D is invertible, i.e.

M(V ) = { f ∈ O X (V ) : f |V \ D ∈ O X (V \ D )∗ }.
Then the following facts hold:
1. M can be given the structure of sheaf of monoids on
X: it is clear that M defines a sheaf, however sum of
invertible functions needs not to be invertible and hence
there is no ring-structure on M(V ) for every V.
∗ . In particular there ex2. M contains the sheaf of units O X
ists a well-defined map M → O X which is the identity∗.
when restricted to O X

3. Informally M “remembers” the inclusion U → X: while
being defined on X encodes information about U andhis complement D.
4. Derivatives of sections of M generate the space of sections of the sheaf of differentials Ω1X ( D ) with logarithmic poles along D which (partially) justifies the terminology of logarithmic geometry.
5. The normal crossing condition can be rephrased in local
terms if one considers the étale topology instead of the
Zariski topology.

6.1.1

Log structure and log-schemes

Motivated by the previous example we give the following
definition due to K. Kato
Definition 6.1.2. Given a scheme X a pre-logarithmic structure on X is a sheaf of monoid M on the étale site of X, together
with a morphism of sheaves of monoids α : M → O X where O X

102

6. Logarithmic Geometryis viewed as a sheaf of monoids under multiplication. The map α issometimes called the structure morphism.
∗ → O ∗ given by α is an isomorphism, M, or
If the map α−1 O X
Xrather the couple (M, α), is called a logarithmic structure on thescheme X.

Remark 6.1.3. 1. The sheaf M defined in example 6.1.1 is alogarithmic structure on the smooth variety X with theinclusion map M → O X .
2. There is an associated logarithmic structure to every prelogarithmic structure (M, α) indicated by M a : is givenby
Ma = O X∗ ⊕α−1 OX∗ M,∗ ⊕M
i.e. the quotient (in the category of monoids) of O X
by the relation on sections over a geometric point P ∈ X
defined by(u · α(v0 ) = u0 · α(v)
0 0
0
−1 ∗
(u, a) ∼ (u , a ) ⇔ ∃v, v ∈ α O X,P :v + a = v0 + a0
∗ is actually(this characterization is possible because O X
a group under multiplication; see [Kat2] (1.3)). Equivalently the associated logarithmic structure is obtainedby the pushout of the following diagram in the categoryof monoids over the étale site∗
α −1 O X

α

/M.



O X∗
The structure map for the associated logarithmic structure M a is given by

Ma → OX(u, a) 7→ u · α( a).

3. Given a monoid M and a ring R there is a canonical logarithmic structure that can be defined on X = Spec R[ M ],

6.1. Background and basic constructions

103

where R[ M] denotes the monoid algebra (the R-algebra
whose underlying module structure is free with basis
M). The log structure comes from the canonical map
M → R[ M ]. Usually the scheme X = Spec R with its
canonical logarithmic structure is denoted bySpec ( M → R[ M]).
Definition 6.1.4. A logarithmic scheme, usually called a log
scheme, is a scheme X together with a log structure (M, α) definedon X. We usually denote the log scheme by X when the logarithmic
structure is clear, or by the couple ( X, M), suppressing the map
α. If we need to specify the underlying scheme rather than the log
scheme we will use the standard notation (after [Ogu] and [ACG+ ])X.
Definition 6.1.5. Given a log scheme ( X, M X ), the characteristic of the log structure M X is the quotient sheaf

M := MX /O X∗ .
In the same way we focus on varieties rather than on arbitrary schemes, we are going to describe a smaller class of logarithmic schemes where the underlying sheaves of monoids
possess nice properties that, quoting Kato, are not “too pathological”. We begin by introducing properties of log structures
analogous to being quasi-coherent, coherent and locally free
for sheaves.
Definition 6.1.6. A log-structure M on a scheme X is called
quasi-coherent if étale locally there exists a monoid P together witha homomorphism P X → O X , where P X denotes the constant sheaf
associated to P, such that the associated logarithmic structure isisomorphic to M. If the monoid P can be chosen to be finitelygenerated then M is called a coherent log structure on X. If P 'Nk for some k, the log structure M is called locally free.
In the same spirit, seeking an analogous for normal schemes,
we give the following definitions:Definition 6.1.7. A monoid ( M, +) is called integral if, forevery m1 , m2 , m ∈ M such that m1 + m = m2 + m it holds m1 =

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6. Logarithmic Geometry

m2 . This is equivalent to require that the natural map M → M gp
is injective, where M gp is the associated group of the monoid M.
A monoid ( M, +) is called saturated if it is integral and for everym ∈ M gp , n ∈ N, if n · m ∈ M then m ∈ M.
Definition 6.1.8. A log structure M is called integral if it isa sheaf of integral monoids. It is called fine if it is coherent andintegral; is called fine and saturated, or more briefly fs, if it is acoherent sheaf of saturated monoids.
Remark 6.1.9. Integral quasi-coherent log structures can becharacterized as follows: given a quasi-coherent log structure
M on a scheme X, M is integral if étale locally is isomorphic to the log structure associated to the pre log structure
P X → O X for some integral monoid P. The same holds forfine log structures replacing quasi-coherent with coherent andintegral with finitely generated and integral (see Definition 6.1.16
for an equivalent reformulation using charts).
Example 6.1.10. 1. In the setting of example 6.1.1 let M D
be the log structure induced by the normal crossing divisor D. By the normal crossing hypothesis, the divisor
D can be written (étale) locally as the union of regularclosed subschemes given by f i = 0 with i = 1, . . . , r forsome positive integer r. In this notation, the log structure M D is the log structure associated to the pre log
structure given by
Nr → O X(n1 , . . . , nr ) 7→ ∏ f ini .
i

In particular M D is fine and saturated, being N finitely
generated, integral and saturated.
2. To an integral monoid P with no invertible element otherthan the unit one can associate an integral logarithmic∗ ⊕P → O
structure M on a scheme X via M := O X
Xdefined by(
u if a = 1(u, a) 7→
0 if a 6= 1

6.1. Background and basic constructions

105

It can be shown that every integral logarithmic structurearise in this way. An important example is the so-called
standard logarithmic point defined in the following way:
let X = Spec k be the 0-dimensional affine scheme fora field k, and consider the integral log structure on X
associated to the integral monoid N as above. Such astructure, with the additive notation for N is given by(
u if n = 0k∗ ⊕ N → OX
(u, n) 7→
0 if n 6= 0
and is usually denoted by(u, n) 7→ u · 0n ,where one defines 00 = 1 and 0n = 0 if n 6= 0.

6.1.2

Log maps

Definition 6.1.11. Given two log structures M1 and M2 on
a scheme X, a morphism of log structure is a morphism of sheavesof monoid M1 → M2 compatible with structure morphisms, i.e.
such that the following diagram is commutative

M1 D

DD
DD
D
α1 DD
"

OX

/ M2 .
z
z
zz
zz α2
z
|
z

Definition 6.1.12. Given a morphism of schemes f : X →Y and a (pre) log structure M on Y, the inverse image of M,
denoted by f ∗ M is the logarithmic structure associated to the prelog structure on X associate to the map
f −1 (M) → f −1 OY → O X .
Example 6.1.13. 1. The canonical log structure defined in
Remark 6.1.3 on Spec R[ M] for a monoid M and a ring R

106

6. Logarithmic Geometry

associated to the map M → R[ M ] can be defined equivalently as the pullback of the canonical log structure on
Spec ( M → Z[ M]) via the map Z → R.
2. The log structure of the standard logarithmic point defined in 6.1.10 can be seen as the inverse image of the log
structure on Spec k[ x1 , . . . , xn ] given by the divisor D :xi = 0 for some i, via the map Spec k → Spec k[ x1 , . . . , xn ]that sends the point Spec k to the origin of the affine
space Spec k[ x1 , . . . , xn ].
With these two notions one can define maps in the category
LSch of log schemes in the following way:
Definition 6.1.14. A morphism of log schemes ( X, M X ) →(Y, MY ) is a couple ( f , f [ ) where f : X → Y is a morphisms in
the category of schemes and f [ : f ∗ MY → M X is a morphism oflog structures on X.
In the same way characteristics were defined for schemes we
can define the analogous notion for maps:
Definition 6.1.15. Given a log morphism f : ( X, M X ) →(Y, MY ) the characteristic of f , or the relative characteristic,is the quotient monoid sheaf M X /N where N is the image off ∗ MY → M X .
In order to work locally on log schemes and log maps, we
introduce the useful notion of chart:
Definition 6.1.16. Given a log scheme ( X, M X ), and a monoid
P, a chart for M X is a morphism P → Γ( X, M X ) (or equivalently P X → M X ) such that the induced map on logarithmic
structure P → M is an isomorphism, where P denotes the log
structure associated to the prelog structure induced by the chart,
i.e. to P → Γ( X, M X ) → Γ( X, O X ).
Remark 6.1.17. Give a map P → Γ( X, M X ) is equivalent togive a map of log schemes( X, MX ) → Spec ( P → Z[ P]),see [Ogu] for more details.

6.1. Background and basic constructions

107

Having defined charts for log structures we can make more
precise Definition 6.1.8 in the following way: given a fine log
scheme ( X, M X ) from the coherence of M X we get a map
P X → O X from the constant sheaf associated to a finitely generated monoid P whose associated log structure is isomorphic
to M X . This map induces a map P → Γ( X, M X ) which is achart for M X . This shows that for fine log schemes charts always exists. In particular fine, fine and saturated and locally
free log schemes can be characterized using charts.
The notion of chart can be extended to morphisms between
log schemes in a natural way:
Definition 6.1.18. Given a morphism of log schemesf : ( X, M) → (Y, N ),a chart for f is a triple(P X → M, QY → N , Q → P)
consisting of
1. a chart P X → M for ( X, M) and a monoid P;
2. a chart QY → N for (Y, N ) and a monoid Q;
3. a monoid homomorphism Q → P.
such that the following diagram is commutative:

QX

/ PX




/M

f ∗N

Remark 6.1.19. Similarly to the case of charts of log structure,
if the schemes ( X, M) and (Y, N ) are fine log schemes, achart for the map f : ( X, M) → (Y, N ) always exists (see[Kat2], Lemma 2.10).

108

6.1.3

6. Logarithmic Geometry

Log smoothness

In the following subsection we are going to discuss the notionof smoothness in the logarithmic category: one of the main
point we are going to focus on, is that log smoothness will
not a priori implies flatness of the underlying morphism. In
this setting the notion of integral log structure will provide
the extra condition needed for such a flatness to hold.
We begin by the following definition:
Definition 6.1.20. Given a morphism of fine log schemes f :( X, MX ) → (Y, MY ), f is called log(arithmically) smooth if
the underlying morphism between schemes f : X → Y is locally offinite presentation, i.e. affine locally represented by a map
Spec A → Spec B

A=

where

B [ x1 , . . . x n ]f1, . . . , fr

and for any commutative diagram of log schemes
T0


φ

j

T1

/X ,


ψ

f

/Y

where j is a (strict) closed immersion defined by a square zero idealand φ, ψ are log morphisms, there exists a morphism g : T1 → Xsuch that φ = g ◦ j and ψ = f ◦ g, i.e. such that the following
diagram commutes
/X .
?

g 
j 
f


 
/Y
T1

T0

φ

ψ

This is equivalent to require the map f to be formally smooth andlocally of finite presentation in the category of log schemes.

6.1. Background and basic constructions

109

Logarithmic smoothness is related to the smoothness of theunderlying map between schemes in the following way:
Proposition 6.1.21. Let f : ( X, M X ) → (Y, MY ) be a log
morphisms between fine log schemes. If f ∗ MY ' M X then f is log
smooth if and only if f is smooth as a map between the underlying
schemes.
Note that a log-smooth map needs not to be smooth as a map
between the underlying schemes. The typical example is themap from a curve with only normal crossing singularities toa log point: one can see that the map is log smooth while theunderlying map between schemes has singularities being the
curve singular. As an explicit example one can consider themap
C[ x1 , x2 ]f : Spec→ Spec (N → C),
x1 · x2
where the first log structure is given by
N2 →C[ x1 , x2 ],x1 · x2

sending ei 7→ xi , and the log map is defined by
NO 2
∆

N

/ C[ x1 , x2 ] ,x1 ·O x2
/Cwith ∆ the diagonal. This is another point in which Log Geometry allows to deal with singularity as if they were smooth.
As mentioned above log smoothness does not imply flatnessof the underlying map but if we require the map to be integral
we get the flatness. Let us first recall the following
Definition 6.1.22. A morphism f : ( X, M X ) → (Y, MY ) offine log schemes is called integral if for every closed point x ∈ X
the map f −1 (MY ) x → M X,x induces a flat map on the monoid
algebras Z[ f −1 (MY ) x ] → Z[M X,x ].

110

6. Logarithmic Geometry

With the integrality assumptions flatness follows from log smoothness. The following theorem holds:
Theorem 6.1.23 (Kato). Given a log smooth and integral morphism between fine logarithmic schemes, the underlying map ofschemes is flat.

6.2

Log stable curves

Logarithmic geometry provides tools to extend the usual notion of stable curve and stable maps to the relative case. For
curves the functor of the moduli problem for genus g, nmarked stable curves is representable by a DM stack Mg,n which compactifies the open stack of smooth irreducible curvesof genus g with n markings; moreover the boundary given by
singular stable curves is a normal crossing divisor D . Therefore it is given a natural logarithmic structure MM on themoduli spaces Mg,n . Analogous construction, using inverse
image of MM and the divisors defined by sections, can bemade on the universal family Cg,n in a canonical way. One
then expects that each stable curve can inherit a logarithmic
structure from these canonical logarithmic structures. This is
the starting point for the definition of log stable curve.

6.2.1

Log curves as stable curves

Recall that a genus g, n pointed pre-stable curve over a scheme
S is a flat projective morphism of schemes C → S with n
sections such that every geometric fiber is a projective, connected, reduced, (at most) nodal curve of arithmetic genus g;
it is called a stable curve if its group of automorphisms is finite. Equivalently, the stability condition can be expressed byrequiring that the twisted dualizing sheaf ωCs ( P1 + · · · + Pn )is ample, where Cs is a geometric fiber and P1 , . . . , Pn are its
marked points. Our first goal is to show that stable genus g n

6.2. Log stable curves

111

pointed curves and log curves are, in some sense, “faces of thesame coin”, i.e. on one side to each genus g, n pointed stable
curve can be given a log structure in a natural way, and on theother side, each log curve has a structure of a usual pre-stable
curve where the log structure encodes information similar tothe pointed structure. These results were first established by
Fumiharu Kato in [Kat1]; we refer to this paper, together with
[ACG+ ], for details and complete proofs.
We begin with the following
Definition 6.2.1 ([Kat1] 1.2). A log curve over an fs log scheme(S, MS ), is a log smooth and integral morphism f : X → S of fs log
schemes such that each geometric fiber is a connected and reduced
curve.
We note that from the integrality assumption it follows thatthe map f is flat as a map between the underlying schemes.
In order to deal with moduli space we specify the definitionof arrows with the following
Definition 6.2.2. An isomorphism of log curves C1 → S andC2 → S over an fs scheme S with log structure MS1 , MS2 respectively, is a couple (σ, γ) where
1. σ : (S, MS1 ) → (S, MS2 ) is an isomorphism of log schemes
such that the underlying map between schemes is the identity;
2. γ : (C1 , M1 ) → (C2 , M2 ) is an isomorphisms of log schemes
whose underlying map between schemes is an isomorphism of
pre-stable curves over S;
3. the two maps are compatible with the log curve structure, i.e.
the following diagram is commutative(C1 , M1 )/ (C2 , M2 )
/ (S, MS )(S, MS1 )2With the definition of arrows we can form a category of log

112

6. Logarithmic Geometry

curves together with a stability condition that we will describe later. The main feature of log curves is that the f s hypothesis together with the log-smoothness restrict the possible singularities that can appear in such curves. The following
theorem holds:
Theorem 6.2.3 ([Kat1] 1.3). Given a log curvef : ( X, M X ) → (S = Spec k, MS )with k separable closed then X has at worst nodal singularities.
Moreover, call r1 , . . . , rl the set of double points of X, then there
exists s1 , . . . , sn points distinct from the nodes, such that the characteristic of f is given by

MX/S = Zr1 ⊕ · · · ⊕ Zrl ⊕ Ns1 ⊕ · · · ⊕ Nsn ,where for a monoid P we denote by Px the skyscraper sheaf supported at x.
As a corollary we see that, not only the log curve structure on( X, MX ) → (Spec k, MS ) gives X a pre-stable structure, but
it gives also a set of “special” points, namely s1 , . . . , sn in theprevious notation; such points should be thought as marked
points for the curve X. Indeed, if every fiber of X → S isa genus g curve, the si make possible to define sections Σi :S → X, and hence recover a genus g, n-pointed structure on
the curve X (see [Kat1], Proposition 1.7 for detailed discussion
and proofs). We will refer to the points si as the marked pointsof the log curve.
We have seen how markings can be recovered from the log
structure on a log curve. Next step is to define the stability
condition in terms of Log Geometry. For this we need thefollowing definition
Definition 6.2.4 (Logarithmic differentials). Given a morphism of log schemes f : X → Y, there exists a O X -module Ω1X/Y ,called the sheaf of logarithmic differentials, which carries a universal derivation (d, dlog) in the logarithmic category and verifies,

6.2. Log stable curves

113

for every O X -module A
HomOX (Ω1X/Y , A) ' DerY ( X, A)
as O X -modules.
In the case Y = Spec (N → k ) is the standard log point, X asmooth variety and M X coming from a normal crossing divisor D, then the module Ω1X/Y is generated by the differentials
with logarithmic poles along D (with a relation). For details
and properties of Ω1X/Y see [Ogu].
Proposition 6.2.5 ([Kat1] 1.13). Given a log curvef : ( X, M X ) → (S = Spec k, MS )with marked points {s1 , . . . , sn }, there exists a natural isomorphismΩ1X/S ' ωX (s1 + · · · + sn ).
From the previous Proposition one can see that the stability
condition can be rephrased in terms of the sheaf of log differentials which naturally encodes the contribution of the markings. Hence we can give the following definition:
Definition 6.2.6 (Log stable curve). Given a log-curve f :X → S let p ∈ S denote a close point; define l = l ( p) and n =n( p) to be the integers such thatMX p /p = Zr1 ⊕ · · · ⊕ Zrl ( p) ⊕ Ns1 ⊕ · · · ⊕ Nsn ( p) .
Then f (X) is called a g, n log stable curve if f is proper, X has
genus g, n( p) = n for every p and Ω1X/S is ample.

6.2.2

Canonical log structure on stable curves

From the previous discussion it follows that every g, n log stable curve is (naturally) a genus g, n-pointed stable curve inusual sense. Next step would be a natural converse of the

114

6. Logarithmic Geometry

previous construction, i.e. given a usual stable curve, define anatural/canonical log structure so that the relative characteristic is supported at the nodes. Since we are going to use thisconstruction in the sequel we briefly recall it.
Let X → S be a stable genus g curve with n markings. We aregoing to restrict to the special case S = Spec k for a field k although the same construction can be done over the spectrumof a strict Henselian ring. Near a node ri of f one can finda Cartesian diagram, over an étale neighborhood of (points
specializing to) ri , that looks as follows:
ψiUi/ Spec k [ x, y, t]( xy − t)
π



ϕi

S



/ Spec k [t]Define on the bottom right corner a log structure N → k [t] by
e 7→ t; in the same way on the upper right corner define a log
structurek [ x, y, t]
N2 →( xy − t)bye1 7 → x

e2 7 → y(e denotes generator for N). Then if ∆ is the diagonal N →N2 , (π, ∆) is a log morphism. Using these structures we canpull them back to S and Ui via ϕi and ψi respectively, obtaining log structures Li and Mi0 .
Away from ri we define Mi00 to be the pullback of Li ; the log
structure Mi on X will be defined as the gluing of Mi0 and
Mi00 . Finally let N be the log structure induced by the divisorof marked points.
Definition 6.2.7 (Canonical log structure on stable curves).
Given a stable curve f : X → S of genus g with n marked points,

6.2. Log stable curves

115

in the previous notations, the canonical log structure on f is thefollowing couple of log structures on X and S respectively

MX = M1 ⊕OX∗ · · · ⊕OX∗ Ml ⊕OX∗ N
MS = L1 ⊕OX∗ · · · ⊕OX∗ Ll .
With this structure f : ( X, M X ) → (S, MS ) is a log curve as
in Definition 6.2.1 where the morphism at the level of log structure
is naturally defined by construction (and by the Cartesian diagram
above).
Remark 6.2.8. The canonical structure on the base MS is locally free, i.e. there exists a non-negative integer l such that
MS,s ' Nl for every closed point s ∈ S. This gives a global
chart Nl → MS . Now the analogous of the local picture described above in the neighborhood of a node, at the level of(pre) log structures looks as follows:M|U o

Nl −1 O⊕ N2

O(id,∆)π[π ∗ (MS )|U o

Nl −1 ⊕ N

The image in MS of the generator of the bottom right N iscalled the element in MS smoothing the node p.
The canonical log structure verifies a minimality condition in
the following sense: given a log curve f : X → S, for any
other log curve f 0 : X 0 → S0 whose underline curve is isomorphic to the base change X 0 ' X ×S S0 and such that thedivisor of marked points of X 0 is sent scheme-theoretically tothe divisor of marked points of X, there exist unique log maps
α, β, extending the maps given by the fiber product, making
the following diagram Cartesian
X0
f0



S0

β

α

/X


/S

f

116

6. Logarithmic Geometry

We end this section by explicitly determining the stalks ofthe characteristic at smooth points, marked points and nodes.
This will give, in particular, a local description of the canonical structure over marked points and nodes in terms of thelogarithmic structure of the base.
Remark 6.2.9. Given a log curve (C , M) with canonical log
structure over a point (Spec k, N ), where N comes from a
monoid P with the unit as the only invertible element, thecharacteristic of M at the level of stalks can be described asfollows:
1. Over a generic point ξ, or a closed point different fromnodes and marked points, the log structure comes entirely from below henceMξ = P.
2. Over a marked point s the log structure comes from thebase plus a contribution of the marked point. In particular, given Theorem 6.2.3 and Definition 6.2.7 one gets

Ms = P ⊕ N.
3. Finally over a node r the log structure has again a component coming from the base and a component depending on the node. From the description given at the beginning of this subsection of the local behavior of thecanonical structure in a neighborhood of a node, one
sees that
Mr = P ⊕N N2 ,where the map N → P, given by 1 7→ p, comes fromC → Spec k and the map N → N2 is the diagonal.

6.3

Log stable maps

In this section we are going to review the extension of thetheory of stable maps to log stable maps via Logarithmic Ge-

6.3. Log stable maps

117

ometry tools. Originated by the work of Kontsevich [Kon] torigorously formulate (and check) prediction in enumerative
geometry, the moduli space of stable maps, compactifying thespace of maps from stable curves to varieties with fixed numerical data, gives the proper framework to build GromovWitten (GW) invariants’ theory and their computation. Since
the works [Li1] and [Li2] (in the algebraic category), relative
GW invariants with respect to a smooth divisor has been extensively studied from different points of view. In a 2001 lecture Bernd Siebert proposed a new approach based on Logarithmic Geometry that, lately, led to establish the right settings for a good moduli theory of log stable maps [GS]. A
different approach was introduced by Chen in [Che1] whichuses marked graphs instead of insights from tropical geometry with some restriction on the target spaces. Both theseapproaches define a suitable condition on the log maps (basicness in Gross and Siebert’s work, minimality in Chen’s paper)
which allows to define moduli spaces for these maps (withsome extra conditions) which are proper and carry a perfectobstruction theory, as well as a virtual fundamental class.
For the purpose of the present work, we are mainly interested
in Chen’s approach, or better in its generalization [AC], since
it deals precisely with the situation we are interested in, i.e.
a target space where the log structure is given by a simple
normal crossing divisor. We refer to the papers [Che1] and[AC] for detailed proofs.

6.3.1

Basic properties of log stable maps

We start by recalling that a stable map f : C → Y is a diagram

C
π



S

f

/Y

118

6. Logarithmic Geometry

with Y a projective scheme, π : C → S a pre-stable curve andf : C → Y a morphism; the stability condition requires thatAutY (Cξ ) is finite for every geometric point ξ ∈ S. When considering moduli problems for stable maps one fixes the genus
g and the number n of marked points of C → S together witha curve class β in the projective scheme Y. Once these numerical data are assigned, there exists a Deligne-Mumford stackMg,n (Y, β) parametrizing stable maps f : C → Y from prestable curves C of genus g with n marked points such thatf ∗ [C] = β. We note briefly that pre-stable maps form a category fibered in grupoids over Sch, the category of schemes,
and that this category is represented by an algebraic stack which contains, as a proper substack, stable maps of fixed numerical data ( g, n, β).
We begin by an extension of this definition to the logarithmic
category.
Definition 6.3.1. A pre-stable log map f : C → Y over a
fs log scheme S is a couple given by a log curve π : C → S anda morphism of log schemes C → Y. It can be represented by thefollowing diagram:(C , MC )πf/ (Y, MY )

(S, MS )where now the diagram lives in the logarithmic category. A prestable log map is called stable if the underlying map between schemes
is stable in the usual sense.
Morphisms of (pre-)stable curves are morphisms of log curves as in
Definition 6.2.2 that are compatible with the log map; more precisely
given two pre-stable log maps f 1 : (C1 , M1 ) → (Y, MY ) and f 2 :(C2 , M2 ) → (Y, MY ) over (S, MS1 ) and (S, MS2 ) respectively,
an isomorphism of pre-stable log maps is an isomorphism of the log-

6.3. Log stable maps

119

curves (σ, γ) such that the following diagram is commutative:-

f1(Y, MY )C(C1 , M1 )
(S, MS1 )γσ/ (C2 , M2 )f2

/ (S, MS )2Previous Definition can be extended to deal with the case offamilies of targets, i.e. replacing the fs log scheme Y with afamily Y → B of projective log schemes (see [Che1] section 2and [AC] section 5). Objects and arrows of the corresponding
category are defined similarly to the previous Definition.
In the same spirit of Remark 6.2.9 we give a local description of a log map in the case where the log structure MY
on the target Y is given by a smooth divisor D. This gives
a map N → MY (see example 6.1.10). This log structure(and more generally every locally free DF-structure (see next
section)) is equivalent to the data of a line bundle L (in thiscase OY (− D )) and a section s ∈ H 0 (L∨ ) which gives a map
L → OY that induces the logarithmic structure on Y.
Remark 6.3.2. Let f : C → Y be a log map over a geometricpoint S = Spec k with log structure MS . Suppose that thelog structure MY on Y is given by a smooth divisor D ⊂ Y
as above and that the log structure of the curve C → S isthe canonical one. For every point p ∈ C in an irreducible
component Z = Z p ⊂ C there is a map at the level of thecharacteristics[f : f ∗ (MY ) p → MC,p .
As in remark 6.2.9 we focus on three cases that exhaust all the[possible behaviors of the map f .
1. Let p be a smooth, non-marked point. Then the map

120

6. Logarithmic Geometry

reads

[

f : f ∗ (MY ) p → MS .

If δ denotes the standard generator of N then we have[that f (δ) = e ∈ MS . Such e is called the degeneracyof the irreducible component Z = Z p . By the description of MY , if f ( p) ∈/ D for all p ∈ Z then the image[of f vanishes. We call a component Z degenerate if its
degeneracy is not zero.
2. Let p be a marked point. Now the maps reads[

f : f ∗ (MY ) p → MS ⊕ N
and then, with the above notation we have that[f (δ) = e + c p · σp ,where c p is a non negative integer and σp denotes thegenerator of N in the stalk of the characteristic of C .
The integer c p is called the contact order of f at p. It canbe identify with the multiplicity of intersection ( f (C) ·D ) f ( p) .
3. Let p be a node, at the intersection of two irreducible
components Z and Z 0 . Let e p be a smoothing of the nodep, and let log x p and log y p be elements in MC givenby the local coordinates of Z and Z 0 respectively, at p.
Then by the construction of the canonical log structureof C → S the following equality holds in MC :e p = log x p + log y p .
[Hence, without loss of generality, the map f can be assumed to be[f (δ) = e + c p · log x p
for a non negative integer c p , called the contact order of fat the node p. The node is called distinguished if c p 6= 0.

6.3. Log stable maps

121

The degeneracy of the two components Z and Z 0 are
respectively e and e + c p · e p . Sometimes Z is called thelower component and Z 0 the upper component of p
The main property of the contact orders is summarized in thefollowing
Lemma 6.3.3. Give a log map f : C → Y over S the following
hold:
1. for every marking Σi , there exists an open subset S0 such thatthe order of contact along Σi is constant;
2. for every node at the intersection of two irreducible component there exists an open subset S00 such that either the nodeis smoothed out or its contact order remains the same.

6.3.2

Deligne Falting pairs and moduli of minimallog stable maps

As for usual stable maps one can construct the moduli spaceassociated to log (stable) maps, which is a category fibered in
grupoids over a log scheme S with fixed genus and marked
points. This stack, denoted by LM g,n (Y ) (or LM g,n (Y/B)
in the relative setting), is a category fibered in grupoids thatassociates to every log scheme (or B-scheme) S the grupoidsof log maps over S, such that the underlying map has source
given by a pre-stable genus g curve with n marked points.
This stack is algebraic in the sense of Artin ([Che1]) but is,
nonetheless, unhandy, containing all possible log structure on
the base S. Since the main goal is to study log maps without
the information on the base structure we will focus on two
open substacks parametrizing a special subclass of log maps
called minimal.
A complete treatment of minimality for log stable maps is beyond the scope of this chapter, nevertheless we gave here thebasic definition and results referring to [Che1] section 3 and[AC] for detailed construction and proofs.

122

6. Logarithmic Geometry

Before moving to minimality we fix the settings in which sucha notion is naturally defined. We begin by the following
Definition 6.3.4. Given a log scheme ( X, M X ) we call it a
Deligne-Faltings pair, often written as DF pair, if M X is locally
free and there is a morphism of locally constant sheaves Nk → M X
which locally lifts to a chart. The map Nk → M X is called a global
presentation of M X and the integer k is called the rank of the
Deligne-Falting pair ( X, M X ).
As a typical example take a smooth Cartier divisor D in asmooth scheme X, and equip X with the log structure M X
coming from D. Then ( X, M X ) is a Deligne-Faltings pair ofrank 1, and M X corresponds to the line bundle O X (− D ) withthe map O X (− D ) → O X .
For the rest of this chapter assume fixed a rank k DF pair(Y, MY ) with Y projective and global presentation of MY
given by Nk → MY . In the case k = 1 to such a pair it corresponds the data of a line bundle L and a section s ∈ H 0 (L∨ )which give a map of sheaves L → OY whose associated log
structure is MY . If s is not a zero section, then, denoted by D
the vanishing locus of s, L = OY (− D ). Moreover if δ is thegenerator of N, which we will identify by abuse of notation
with its image in MY , we have that δ lifts, locally, to a sectionof OY whose vanishing locus gives the divisor D.
If k > 1 then for i = 1, . . . , k one has maps αi : N → Nk givenby the inclusion of the i-th component. The composition
N

αi

/ Nk

/ MY

defines a rank 1 DF structure MY,i on Y which gives a decomposition

MY =

k
M

MY,i

i =1

in which each MY,i , by the discussion above, corresponds toa couple (Li , si ) of line bundles and sections. As an example
of such a construction, generalizing the above example, one

6.3. Log stable maps

123

can consider a simple normal crossing divisor in which each
Li corresponds to O X (− Di ), for an irreducible component Di
of D.
The first definition of minimality follows from the log structure of the moduli space in the following way: consider the
category LM g,n (Y ) fibered over fs log scheme, instead of log
schemes. Such a category admits a fs logarithmic structure
as an Artin stack; denote it by K(Y ) = (K, MK ): stable log
maps over S are equivalent to log morphism S → K(Y ). The
existence of such a fs log stack permits to identify a minimal
structure on log maps over usual schemes by pulling back thelog structure MK .
More formally, given a scheme S, an object of the stack K(Y )is an arrow S → K(Y ). This map gives a log scheme S = Smin
where the log structure is pulled back from K(Y ). Now fromevery log map over S, f : C → Y, one obtains a log map
with log structure pulled back from K(Y ) and its universal
family C. This can be made explicit in the following Cartesian
diagram:C/ C min/C/ Smin/ K(Y )S/ K(Y )

SA

AA
AA
AA
AA 

/Y

Definition 6.3.5. A log map f : C → Y over S is called minimal if the log structure on C → S is pulled back from the log structure of K(Y ). This is equivalent to require that the map S → K(Y )corresponding to f is strict.
Minimality can be checked combinatorially from the map f :C → Y without any reference to the log structure of the stackK(Y ). This has the advantage that, given any log map, one
can define explicitly a minimal log structure associated to f .
In fact one can do better and obtaining the following:124

6. Logarithmic Geometry

Theorem 6.3.6 (Q. Chen). To any log map f : (C , MC ) → Y
over an fs log scheme (S, MS ) there exists a minimal log map over
(S, MS min ), and a map of fs log schemesΦ : (S, MS ) → (S, MS min ),both unique up to isomorphism, such that the following diagram iscommutative-f(Y, MY )B(C , MC )

(S, MS )ΦCΦ/ (C , MC min )

f min


/ (S, MS min )Construction 6.3.7. We briefly sketch the ideas underlying Chen’s
combinatorial description of minimality for the case k = 1. Toeach log map f : C → Y over S there is a canonical graph associated, namely the dual graph of the curve C . To such a graph,
Chen gives weights to the edges given by contact orders andan orientation to a subset of them coming from the partial
order defined by upper and lower components; he then constructs a monoid M( G f ) which posses a generator for eachvertex and edge together with some natural relations comingfrom the local behavior of the map at the level of characteristic
described in subsection 6.3.1. By quotienting out the torsion
part of the associated group and taking the saturation of theimage of the monoid M( G f ) in such a quotient one obtains a
sharp monoid, M( G f ) with a canonical map
φ : M( G f ) → MS .
Then the map f is minimal if and only if φ is an isomorphism.
We now fix discrete data that gives a partition of the moduli
space of log stable maps.

6.3. Log stable maps

125

Notation 6.3.8. Denote by Γ the fourple ( β, g, n, ~c) where
1. β ∈ H 2 (Y, Z) is a curve class;2. g is a non-negative integer denoting the genus;
3. n is a non-negative integer denoting the number of marked
points;
4. ~c is a set of tuples, denoting the contact order of themarkings. In particular ~c = (cij ) with i = 1, . . . , k andj = 1, . . . , n such thatnβ · c1 ( Li∨ ) = ∑ cij .
j =1Here the Li are the line bundles associated to the DF pair(Y, MY ) of rank k, and c1 denote the first Chern class.
Definition 6.3.9. A minimal log stable map f : C → Y overa geometric point S is called minimal with respect to Γ, or Γminimal if
1. the curve C → S is a log pre-stable curve of genus g withn-markings;
2. the pushforward of the fundamental class of C equals β, i.e.
f ∗ [C] = β;
3. the contact order along the i-th marking with the j-th component D j of the divisor D is given by cij .
The stack parametrizing minimal stable maps with genus g, nmarkings and curve class β is denote by K g,n (Y, β) and the stackparametrizing Γ-minimal stable maps is denoted by KΓ (Y ).
The above stacks are both algebraic and the following partition, of open and close substacks holds:

K g,n (Y, β) =GΓK Γ (Y ) .

126

6. Logarithmic Geometry

The main properties of the above stacks are summarized in
the following
Theorem 6.3.10 (Q. Chen, Abramovich - Chen). The fiberedcategory KΓ (Y ) is a proper Deligne-Mumford stack with a representable and finite map to the stack of standard stable maps obtainedby removing log structures.
Remark 6.3.11. The projectivity of the scheme Y is essential
for the properness of the stack; however some properties canbe recovered with the weaker hypothesis of Y being separated
and of finite type over C.

7Complements of a very
generic quartic
The main goal of this chapter is to extend Theorem 3.2.3 tocomplements of very generic quartics. The main idea is toconsider flat deformations of degree four divisors in P2 andreformulate Corvaja and Zannier result in order to make itinvariant under such deformations of the boundary divisor.
The tool used here is Logarithmic Geometry in the sense of
Kato and Illusie as introduced in Chapter 6. We will begin by
an extension of the result for complements of a conic and twolines to log-stable maps. The aim is to give a “moduli” reformulation relating the degree bound to a vanishing of certain
moduli space of log stable maps to DF pair as defined by Chen
and Abramovich-Chen (see Section 6.3). This will be the focusof the second section where we will prove that the strengthen
version of Theorem 3.2.3 is actually equivalent to the emptiness of some stacks of (minimal) log stable maps. Once thisequivalence is settled we use the properness of the stacks toobtain the same vanishing for stacks constructed from complements of more general degree four divisor.

128

7.1

7. Complements of a very generic quartic

Three components case for log-stable maps

Corvaja and Zannier prove Lang-Vojta Conjecture in the split
function field case for the complement of a conic and two lines
in P2 as described in detail in section 3.2.1 at page 50. Theirmain result, i.e. Theorem 3.2.3 page 50 states that morphisms
f : C˜ → P2 such that C˜ is a smooth projective curve, f −1 ( D ) ⊂S for a divisor D in P2 consisting of a conic and two lines withnormal crossing and a finite set of points S ⊂ C˜, verify
˜ ≤ 215 · 35 · max{1, χ(C˜ \ S)},deg( f (C))
where as usual the Euler characteristic of the affine curve C˜ \ S
is defined as
˜ = 2g(C)˜ − 2 + #S.
χ(C˜ \ S) := χS (C)The first step towards a generalization of Theorem 3.2.3 is toconsider more general maps than the one considered by Corvaja and Zannier. The reason is that we want to use deformation arguments applied to our situation and this led naturally
to consider reducible curves.

7.1.1

Extending Corvaja and Zannier Theorem

Consider a flat deformation Dt of the divisor D, a conic andtwo lines in general position, in which the general member
is a quartic and the special fiber over a geometric point is
D0 = D. Construct the associated fibered threefold X in
which every fiber is P2 with the corresponding divisor Dt .
We want to study families of curves that moves in the deformation, i.e. such that when restricted to a fiber give a plane
curve. In particular we want to answer the following questions:
Q1: Suppose that the image in the general fiber is smooth:what can be sad about the image in the special fiber
P2 , D0 ?

7.1. Three components case for log-stable maps

129

A: From the theory of stable map we know that such a curve
could acquire at most nodal singularities and in particular can become reducible.
Q2: What can be said about the variation of the number ofpoints of intersection with the divisor Dt ?
A: The question here is more subtle: the main difficulty isthat multiplicities of intersection cannot be controlled
a priori in the family. Even worse one of the components of the curve can be mapped on a component of
the boundary divisor. In such a case even defining themultiplicity of intersection can be troublesome. That isprecisely the point in which Logarithmic Geometry willplay an important role.
From the previous discussion there is some evidence that wehave to deal with morphisms from pre-stable curves, i.e. reducible connected curves with at worst nodes as singularities. The first thing we have to care about is the behavior of
the set S, i.e. the set of points in the curves that are mapped
to the divisor, when considering reducible curves instead ofirreducible ones. With this goal in mind we first define theconcept of partial normalization of a curve.
Definition 7.1.1. Let C be an affine, reducible and at worst
nodal curve. We resolve the nodal singularities which do not occur
as points of intersection of different components. The strict transform of C gives rise to a curve of the form C˜ \ S for a projective,
reducible and at worst nodal curve C˜ in which every node occurs
as a point of intersection of two different components; we call C˜ \ S
a partial normalization of C . In particular all partial normalized
curves have simple normal crossing singularities.
As an example consider the figure 7.1 in which a blow-up
of a curve C˜ at the node P is pictured: the blow-up resolves
singularities in the irreducible components distinct from thenormal crossing singularities arising as intersection between
different components.

130

7. Complements of a very generic quartic

bl P

/

Q1
P1

Q1

Q2

Q2

P

P2
Q3

Q3

Q4

Q4

Figure 7.1: Partial normalization of a curve
Now from a partial normalization of C we would like to consider the set S such that it is additive with respect to the irreducible components of C , i.e. if C1 , . . . , Cr are the irreducible
components of the partial normalization of the pre-stable
curve C and Si denotes the set of points of S that lie in theirreducible component Ci , we want to understand whether isit true thatr

∑ #Si = #S.
?

i =1

It is easy to construct examples in which the former equality
does not hold. Consider a two component plane curve C1 ∪ C2
in which each component is isomorphic to a line and theyintersect in a unique node P = C1 ∩ C2 . Suppose that the only
point of S in the curve is the node. Then one has#S1 = 1#S2 = 1

#S = 1.

As this example shows, the problems can arise when considering curves in which the points of S are nodes. In facts
nodes are special points of two different irreducible components and therefore they count twice when considered in eachcomponent separately, but at the same time, they are counted
once in the total curve. More in general one can prove that
if the set S does not contain any node then its cardinality isadditive in the previous sense. On the other hand, when the7.1. Three components case for log-stable maps

131

set S contains nodes, the additivity fails to hold. Neverthelessone can relate the total number of points of the set S with thesum of the number of points in each irreducible component
in the following way:
Lemma 7.1.2. Let C be the partial normalization of a pre-stable
curve and let S as before. Let Ci be the i-th irreducible component of
C for i = 1, . . . , r and let Si = S ∩ Ci , i.e. the set of points of S in
the irreducible component Ci , then the following holds:r∑ #Si ≤ #S + # nodes .

i +1

Proof. For previous discussion, if S does not contain any node
then #S is additive in its components and hence the conclusion holds trivially. Suppose that S contains exactly a node P:
then such a node is a point of at most two components, say Ci
and C j . This implies thatSi ∪ S j = (Si \ { P}) ∪ (S j \ { P}) ∪ { P}and then, since#Si + #S j = #S ∩ (Ci ∪ C j ) + 1,
one getsr∑ #Sk = #S + 1.

k =1

From the previous calculation we get that if S contains n nodes
then sum of the number of points of S in each component isprecisely equal to #S + n. Hence, in the worst situation inwhich S contains all the nodes we get the upper bound required.
Remark 7.1.3. From the proof of Lemma 7.1.2 we actually
proved something more: we showed that if n is the numberof nodes contained in S then#S = ∑ #Si − n,132

7. Complements of a very generic quartic

which in particular implies that the cardinality of S is additive
if no nodes are contained in S. However, for future needs, wewill need only the estimate given in the previous Lemma.
Next step is to study how the Euler characteristic behave for
reducible curves with respect to the Euler Characteristic of
the irreducible components. We split the description in thefollowing two lemmas, treating first the case in which all theirreducible components have positive Euler characteristic.
Lemma 7.1.4. Let C˜ = ∪in=1 Ci (n ≥ 2) be a projective, reducible and at worst nodal curve, in which every node occurs as
˜ ,a point of intersection of two different components; let g = g(C)˜
gi = g(Ci ) and let S be a finite set of points on C with Si = S ∩ Ci .
If χSi (Ci ) > 0 for all i, then the following holds:n∑ max{1, 2gi − 2 + #Si } ≤ 2g − 2 + #S

i.e.

i =1
n

˜ .
∑ max{1, χS (Ci )} ≤ χs (C)ii =1˜ ≥ 1.
In particular χS (C)Proof. The starting point is the formula for the genus of a connected at worst nodal projective curve which reads as follows:n˜ = ∑ g(Ci ) + #nodes − #components + 1.
g(C)

(7.1)

i =1

From this and from Lemma 7.1.2 we get that˜ = 2g − 2 + #S
χS (C)n= 2 ∑ gi + 2# nodes − 2# comp + #S
i =1n

≥ 2 ∑ gi + 2# nodes − 2# comp + ∑ #Si − # nodes
i =1n≥ ∑ χSi (Ci ) + # nodes .
i =1

7.1. Three components case for log-stable maps

133

Now, we are assuming that for every component χSi (Ci ) ≥ 1.
Thenn˜ ≥ ∑ χS (Ci ) + # nodes
χS (C)ii =1n≥ ∑ max{1, χSi (Ci )}.
i =1Previous Lemma leaves out the case in which irreducible components of the curve have non positive Euler Characteristic.
From geometric construction one can see that the only casethat do not follow in the previous result is the one in which
one of the components has genus 0 and only two points of
S. In this case one cannot recover the same result but only a
optimal estimate that read as follows:
Lemma 7.1.5. With the notation above, without any restriction on the Euler characteristic of the irreducible components thefollowing holds:n˜ ≥ ∑ χS (Ci ) + # nodes .
χS (C)ii =1

In particular, every time there exist more than one irreducible com˜ is always positive.
ponent, the Euler characteristic χS (C)Proof. The desired bound follows from Lemma 7.1.4 together
with the lower bound:# nodes ≥ # components − 1.

7.1.2

Extension to log stable maps

As mentioned at the beginning of this section we want toextend Theorem 3.2.3 (page 50) to a more general class of

134

7. Complements of a very generic quartic

maps. This class of maps will be the class of logarithmic stablemaps introduced in Section 6.3 Chapter 7. This choice is natural in the following sense: maps considered by Corvaja andZannier were morphism from abstract nonsingular curves C˜with a distinguished set of points S (whose complement corresponded to the normalization of an affine curve) to P2 , suchthat the inverse image of the intersection between the (imageof) the curve and the divisor D (a conic and two lines in general position) was precisely S. In this setting the target (P2 , D )of such maps is a Deligne-Falting pair, being D a simple normal crossing divisor. Moreover the divisor S on the source
curve gives a log-structure over C˜, and extends the map to alogarithmic map being S precisely the inverse image of D. In
particular the map between log structures is strict and the log
map is automatically stable, because we require the source tobe irreducible, non singular and the map to be non-constant.
For this reasons, having in mind a moduli construction, wewant to study whether the conclusion of Corvaja and Zannier’s Theorem holds more in general for a generic stable logmap (C˜, S) → (P2 , D ).
We are then concerned with log-morphisms ϕ : C˜ → P2 such˜ ∩ D are conthat the pre-image of points of intersection ϕ(C)tained in S. For our needs being a stable log-morphism fromthe curve C˜ to P2 with respect to the divisor D is equivalent
to the following conditions:
1. the underlying map between schemes is stable;
2. For every irreducible component Ci of C˜ such that Ci
maps to a degree one irreducible component of D, Si
contains at least three points.
3. For every irreducible component C j of C˜ such that C jmaps to the degree two irreducible component of D, S jcontains at least four points.
Here Si ⊂ S is again the set of points of S lying in the irreducible component Ci , i.e. Si = S ∩ Ci . Then extension of7.1. Three components case for log-stable maps

135

Theorem 3.2.3 follows from the following theorem (from now
on, in order to simplify the notation we put A = 215 · 35).
Theorem 7.1.6. Given C˜, S, D as above, let ϕ : C˜ → P2 bea non-constant log-morphism such that ϕ−1 ( D ) ⊂ S. Then the
˜ verifies:
degree of the image ϕ(C)˜ ≤ A · max{1, χ(C˜ \ S)}.
deg( ϕ(C))
Proof. If C˜ is irreducible the conclusion is given by Theorem
3.2.3. Hence we will assume that C˜ has more than one irreducible component. From Lemma 7.1.5 this implies that the
˜ is strictly positive.
Euler Characteristic χS (C)We divide the proof in three steps:
Step 0 We begin by proving the theorem for the case in whichC˜ has precisely two irreducible components C˜ = C1 + C2
such that ϕ(C1 ) = D1 = {line}, and ϕ(C2 ) meets properly the divisor D with χS2 (C2 ) > 0. This will illustrate
the main ideas of the proof in a handy case, before considering the general situation. Denote as usual S1 , S2
as the set of points of S in the two irreducible components C1 and C2 respectively. In this case, being ϕ alog-morphism, we have #S1 ≥ 3. Define f i := ϕ|Ci for
i = 1, 2; in this setting we obtain
˜ = deg f 1 (C1 ) + deg f 2 (C2 )deg ϕ(C)= 1 + deg f 2 (C2 ).
We turn our attention to the component C2 : the imageof this smooth curve has proper intersection with thedivisor D and hence falls into the cases of Theorem 3.2.3.
Thus
deg f 2 (C2 ) ≤ A · max{1, 2g(C2 ) − 2 + #S2 }.
At the same time we have Equation 7.1 for the genus ofa connected at worst nodal curve, from which, together

136

7. Complements of a very generic quartic

with the stability condition, we get
˜ = deg f 1 (C1 ) + deg f 2 (C2 )deg ϕ(C)≤ 1 + A max{1, 2g(C2 ) − 2 + #S2 }≤ A[2g(C1 ) − 2 + 3] + A max{1, 2g(C2 ) − 2 + #S2 }≤ A[2g(C1 ) − 2 + #S1 ] + A max{1, 2g(C2 ) − 2 + #S2 }≤ A[2g(C1 ) − 2 + #S1 ] + A[2g(C2 ) − 2 + #S2 ]≤ A 2[ g(C1 ) + g(C2 )] − 2 · 2 + #S1 + #S2 .
Using Lemma 7.1.2, which gives #S1 + #S2 bounded above
by #S plus the number of nodes, and the formula for the
genus of a pre-stable curve we conclude that˜ ≤ A 2[ g(C1 ) + g(C2 )] − 2# comp + #S + # nodes
deg ϕ(C)≤ A 2[ g(C1 ) + g(C2 )] − 2# comp + 2# nodes + #S
˜ − 2 + #S}≤ A 2g(C)˜ − 2 + #S}.
≤ A max{1, 2g(C)We observe that, as in the proof of lemma (7.1.4) we canalways assume #S2 ≥ 2; moreover the same proof could
be applied in the case of a two-component curve in which one of the two components maps over the degreetwo component of the divisor D.
Step 1 Having Step 0 we apply the same ideas to a more general situation. We start from a connected at worst nodal
affine curve whose partial normalization, in the sense
defined above, is C˜ \ S where C˜ is a connected at worst
nodal projective curve in which the nodes occur only
as intersection of two different components. For a fixed
log-morphism ϕ : C˜ → P2 such that ϕ−1 ( D ) ⊂ S, wearrange the components of C˜ in four groups so thatC˜ = t Ci t G j t Dh t Ek ,7.1. Three components case for log-stable maps

137

where the image of the components Ci meets properly
the divisor D and χSi (Ci ) > 0, the image of the components G j meets properly the divisor D and χSj (Ci ) = 0(i.e. curves of genus 0 with two points in each S j ), ϕ(Dh )overlaps a line of the divisor D and Ek is mapped overthe degree-two component of D. We note that one canprove that in the three components case the only images of curves with zero Euler characteristic are lines; inparticular their degree is always strictly less than 2. Asbefore, for every component Ci we can apply Theorem(3.2.3) which gives

deg ϕ(Ci ) ≤ A · max{1, 2g(Ci ) − 2 + #Si }.

Where Si is the set of points of S lying on Ci . For the sets
S and Si we have the following estimate#S ≥∑ #Si + ∑ 2 + ∑ 3 + ∑ 4 − # nodes .

i:Ci ∈C˜

j:G j ∈C˜

h:Dh ∈C˜

k:Ek ∈C˜

As before we want to relate the genus of the curve C˜ to
the genus of its irreducible components; Equation 7.1 inthis setting gives
g(C) = ∑ g(A) + # nodes − # components + 1
A∈C˜

where the sum runs over all the irreducible componentsof C˜. Note however that all the components different
from the Ci have genus zero and hence do not contributeto the sum. With these two inequalities we can analyze

138

7. Complements of a very generic quartic

the degree of the image of the curve obtaining that˜ = ∑ deg ϕ(Ci ) + ∑ deg ϕ(G j )+deg ϕ(C)ij+ ∑ deg ϕ(Dh ) + ∑ deg ϕ(Ek )hk≤ ∑ A max{1, 2g(Ci ) − 2 + #Si } + ∑ 2 + ∑ 3 + ∑ 4
i

j

h

k

≤ A{2[∑ g(Ci ) + ∑ g(G j ) + ∑ g(Dh ) + ∑ g(Ek )]+
i

j

h

k

− 2# comp + ∑ #Si + ∑ 2 + ∑ 3 + ∑ 4}
i

j

h

k

≤ A{2[∑ g(Ci ) + ∑ g(G j ) + ∑ g(Dh ) + ∑ g(Ek )]+
i

j

h

k

− 2# comp + ∑ #Si + ∑ #S j + ∑ #Sh + ∑ #Sk }
i

j

h

k

≤ A{2[∑ g(Ci ) + ∑ g(G j ) + ∑ g(Dh ) + ∑ g(Ek )]+
i

j

h

k

− 2# comp + #S + # nodes }≤ A{2[∑ g(Ci ) + ∑ g(G j ) + ∑ g(Dh ) + ∑ g(Ek )]
i

j

h

k

− 2# comp + #S + 2# nodes }˜ − 2 + #S}≤ A{2g(C)= A max{1, χ(C˜ \ S)},
where the last equality follows from the assumption thatC˜ has more than one irreducible component and from
Lemma 7.1.5.
Step 2 Finally we prove the Theorem for a generic log stable
map ϕ : C˜ → P2 with ϕ−1 ( D ) ⊂ S. This time we divide
the irreducible components of C˜ in two groupsC˜ = tCi t P j ,where P j are the components contracted to points by
the map ϕ. From Step 1 we know that the restriction

7.2. Emptiness of moduli spaces

139

of ϕ to the irreducible components not contracted by ϕverifies the conclusion of the Theorem, i.e., called C theunion of all such components and f = ϕ|C we havedeg f (C) ≤ A · max{1, χ(C \ SC )},with SC = S ∩ C , provided that such a union of components is connected. If it is not connected is enough toconsider each connected component separately. At thesame time the degree of the image of C˜ through ϕ is thesame as deg f (C). Therefore the conclusion holds by thefollowing computation:
˜ = 2g − 2 + #S ≥ 2g(C) − 2 + #SC ,χS (C)
being the genus not less than the sum of the genus of its
irreducible components, and being SC contained in S.

Previous Theorem prove that the same result as Theorem 3.2.3
holds more generally for log-stable maps. In particular, image of log-stable maps is bounded by the Euler Characteristic.
Moreover, it shows that the stability conditions gives a boundfrom below to the number of points of S in a component thatmaps over an irreducible component of the divisor D. At thesame time a bound for the S ∩ P j is given, in such a way thatfor every contracted component Pj its Euler Characteristic is
positive, for every genus g( Pj ).

7.2

Emptiness of moduli spaces

Our ultimate goal, as discussed in Chapter 5, is trying to extend these results to complements of divisors in P2 with lessthan three irreducible components. The idea we would liketo formalize is the following:

140

Dt1

7. Complements of a very generic quartic

Ct1

Dt2

Ct2

Dt0

Ct0

Figure 7.2: Idea of extension to lesser components.
Let D be a quartic in P2 which flattly deform to a conic andtwo lines. Consider such a deformation as a flat fibration overa connected base curve. If every curve in the fiber of D moves
in the family it gives rise to a curve in the fiber of the conic
and two lines. Such a curve should verify the bound of Theorem 7.1.6. If the curve has been moved keeping track of theintersection multiplicities with the divisors in each fiber thena similar bound holds also with respect to the divisor D in thestarting fiber.
A picture of the previous idea can be seen in figure 7.2: here
the algebraic family of divisors Dt is considered over an affine
open of P1 and three fibers are drawn: Dt1 , Dt2 are quartics
having two irreducible components while Dts is the conic andtwo lines appearing in Theorem 7.1.6.
From figure 7.2 one can see how the cardinality of the set S cana priori change in a deformation of the divisor D. We have already seen how to define intersection with D for stable curves
with components mapping to an irreducible component of D:log stability condition gives a lower bound for the point of
S in such irreducible component, even if the component ismapped over an irreducible component or it is contracted by
the map. Now we want a way to control the set S with respect to a flat family of divisor. This in particular will cover
the case of a two conic deforming to a conic and two lines, oreven more generally a quartic deforming to a conic and twolines.

7.2. Emptiness of moduli spaces

141

Instead of proving that #S can be controlled in a flat deformation we will prove that, given numerical data Γ, any classof log-stable maps with such data has constant #S. Then we
will see how properties of certain moduli space for maps with
fixed numerical data Γ are related to Theorem 7.1.6.

7.2.1

Algebraic Hyperbolicity via moduli of log-stablemaps

Let us look more carefully to the inequality of Theorem 7.1.6:
˜ ≤ A · max{1, χS (C)}˜ .
deg f (C)˜
If we denote by β the curve class in P2 corresponding to f (C)˜
we have that deg f (C) = deg β. Then if g and n denotes thegenus of C˜ and the number of points of S we have that previous inequality can be rewritten as
deg β ≤ A · max{1, 2g − 2 + n};
in particular this implies that we cannot choose independently
the quantities g, n and β for a map f . The first consequence of
this reformulation is the following
Proposition 7.2.1. Let (P2 , M D ) be the DF pair given by considering the log structure induced by D. Let Γ = ( g, n, β, ~c) be
discrete data such thatdeg β ≥ A · max{1, 2g − 2 + n}.
Then the moduli space KΓ (P2 , M D ) is empty.
Proof. Every element of KΓ (P2 , M D ) is a minimal log stablemap f : C˜ → P2 . By the description of Γ given in 6.3.8 at page
125, sincen
β · c1 OP2 ( Di ) = ∑ cij ,
j =1

142

7. Complements of a very generic quartic

with D = D1 + D2 + D3 as usual, we see that #S < n (sincesome of the cij may a priori be zero). In these settings, Theorem 7.1.6 gives
˜ ≤ A · max{1, 2g − 2 + #S}
deg β = deg f (C)≤ A · max{1, 2g − 2 + n}.

In the spirit of previous Proposition we give the following
Definition 7.2.2. In the settings above we call a discrete data
Γ = ( g, n, β, ~c) admissible ifdeg β ≤ A · max{1, 2g − 2 + n}.
If ~c as at least one element non zero for each j, we call Γ strictly
admissible. We say that Γ is A-admissible if we want to explicitly
specify the dependency from the constant A.
One of the implication of the Proposition is that, given discrete data Γ, all the Γ-minimal log stable map in KΓ (P2 , M D )satisfy #S ≤ n independently of ~c. Moreover, for every vector of multiplicities ~c the set S is precisely the set of marked
point Pα for which at least one of the ciα is non zero. In particular, fixing Γ will automatically fix also #S for all the maps inKΓ (P2 , M D ). We can rephrase this in the following
Corollary 7.2.3. Theorem 7.1.6 implies that KΓ (P2 , M D ) is
empty for every vector of multiplicities ~c in Γ, for every non admissible discrete data Γ.
Note that Proposition 7.2.1 and Corollary 7.2.3 are stated in
the case in which D has three components however one could
give a more general result, i.e. algebraic hyperbolicity of P2 \D for a simple normal crossing divisor D of any degree andany number of irreducible components implies the emptinessof the corresponding moduli spaces of log stable maps.
Now the deformation argument we sketched before can beapplied in the following way: the properness of the stack7.2. Emptiness of moduli spaces

143

KΓ (P2 , M D ) implies that the equality KΓ (P2 , M D ) = ∅ forall non admissible Γ extends to the general fiber of a flat deformation of the divisor D. Therefore, in order to prove algebraic hyperbolicity for the complement of a generic quartic in
P2 we only need to show that the emptiness of KΓ (P2 , M D )for non admissible discrete data implies algebraic hyperbolicity.
Proposition 7.2.4. Let B > 0 be an integer. Given a simple
normal crossing divisor D in P2 denote by M D the log structureassociated. If for any discrete data Γ which is not B-admissible themoduli space KΓ (P2 , M D ) is empty then every non-constant morphism f˜ : C˜ → P2 verifies:
˜ ≤ B · max{1, χS (C)}˜ ,deg f˜(C)where S = f −1 ( D ).
Proof. Let f˜ : C˜ → P2 , be a non-constant morphism. From
Theorem 6.3.6 at page 124 it follows that there exists a minimal log map f over a geometric point having as underlying
map f . To such a minimal map it corresponds a point in a
˜ , n, f ∗ [C]˜ , ~c).
moduli space KΓ (P2 , M D ) for some Γ = ( g(C)Note that n here comes from the data associated to the prestable curve naturally defined by f˜, i.e. it does not depend on
the log structure associated. In particular, if we fix the set ofmarked points as the set of pre-images of points of intersec˜ · D, it follows that n = #S. Now the existence oftions f˜(C)the map f implies that KΓ (P2 , M D ) is not empty. Therefore,by hypothesis Γ is B-admissible. Then
˜ = deg β ≤ B · max{1, 2g − 2 + n}
deg f˜(C)= B · max{1, 2g − 2 + #S}.

Given Proposition 7.2.1 and Proposition 7.2.4 one can state thefollowing

144

7. Complements of a very generic quartic

Theorem 7.2.5. Let D be a simple normal crossing divisor in2
D be the log structure associated. Then P \ D is
algebraically hyperbolic if and only if there exists a positive constantA such that for every discrete data Γ not A-admissible the moduli
space KΓ (P2 , M D ) is empty.
P2 and let M

7.3

Properness of the stack

Theorem 7.2.5 gives a sort of “moduli interpretation” of algebraic hyperbolicity, translating the problem to the emptinessof some moduli space of logarithmic stable maps. In particular in the three components case, Corvaja and Zannier result
together with its generalization, Theorem 7.1.6, can be readas a property of such moduli spaces, providing emptiness for
non A-admissible discrete data with constant A = 215 · 35.
Moreover, the emptiness of the moduli space is a property
that behaves well with respect to flat deformation of the divisor D. In particular, the following property of KΓ (P2 , M D )holds:
Proposition 7.3.1. Let D, M D as before. Then there is a propermap

KΓ (P2 , M D ) → M ( D ),where M ( D ) is the moduli of divisor D in P2 of fixed degree.
Previous proposition follows from Theorem 6.3.10 and thefact that D here is assumed (simple) normal crossing, hence inparticular toroidal. Now giving a deformation of the divisor
D (3) = D1 + D2 + D3 one gets that the properness of the map
allows to extend the emptiness of the stack KΓ (P2 , M D(3) ) to
the generic member of the deformation. In particular thisgives emptiness of the corresponding stack for the genericquartic degenerating to the divisor consisting of a conic andtwo lines. The union of all such conditions over all possible genus, number of marked points and degree of the pushforward of the fundamental class of a curve, gives algebraic

7.3. Properness of the stack145

hyperbolicity for the complement of a very generic quartic.
Therefore the following theorem holds:
Theorem 7.3.2. Let D be a very generic quartic in P2 . If D has
only simple normal crossing singularities then P2 \ D is (weakly)
algebraic hyperbolic.
Here by very generic we mean a part from countably many
exception, i.e. in the complement of a countable union ofproper closed subvarieties in the Hilbert scheme of plane
quartics, Hilb2 (4t − 2) = Proj( H 0 (P2 , OP2 (4))) . Note thatthe only interesting part of previous Theorem is given by thecases of log irregularity strictly smaller than two; the othercases follows from Theorem 3.1.1 and Theorem 3.2.3. Moreover the constant appearing in the hyperbolicity condition
here is the same as the one given by Corvaja and Zannier
Theorem. in particular if a different strategy were considered,i.e. extending the four lines’ case instead of the three components’ one, the constant could have been lowered also for thegeneric quartic with three components. The rest of the chapter will be devoted to the proof of Theorem 7.3.2.
We stress that the very generic hypothesis will allow to drop
the simple normal crossing requirement, since a very genericquartic can be always be chosen to have normal crossing singularities. However we still prefer to explicitly state it because the result on the properness of the stack requires it as afundamental hypothesis. Present work by Abramovich, Chen
and others seems to give the possibility to obtain the properness of the stack dropping this hypothesis and then making it
redundant also in our context.
The main step in the proof of Theorem 7.3.2 is to prove thatbeing algebraic hyperbolic does not depend on the numberof irreducible components of the divisor D. Once such independence is proved the result will follow from the extensionsof Corvaja and Zannier’s Theorem 3.2.3. This invariance will
be proved using the moduli interpretation of hyperbolicity
given by Theorem 7.2.5 and using the fact that for the generic146

7. Complements of a very generic quartic

member of a deformation to a conic and two lines the stackKΓ (P2 , M D ) coincide with the same stack in the special fiber.
Hence the emptiness of these moduli spaces in the case of
three components extends to the case of very generic quartic. Then the equivalence between the emptiness of the stacks
for non admissible discrete data and algebraic hyperbolicity
will finish the proof.

7.3.1

Invariance under deformation

In this subsection we focus on the following
Proposition 7.3.3. Let D be the generic member of a flat deformation of a quartic into D (3) , a divisor consisting of a conic andtwo lines. Then for any discrete data Γ one has

KΓ (P2 , M D ) = KΓ (P2 , M D(3) ),where M D and M D(3) are the log structure associated to D and
D (3) respectively.
Proof. The only real issue in the previous statement is thata flat deformation of a quartic is not in general log smooth.
Thus the total space in the log category does not automatically give a log map which allows to use the proper map of
Proposition 7.3.1. However, since all the log structures involved come from normal crossing divisor, it is enough toconsider a normal crossing divisor in the total space such that,its restriction to the special fiber gives D (3) , and the restriction
to the generic fiber gives D.
Consider a flat deformation h : D → B of quartics such that
Db = D for the generic point b ∈ B, and Ds = D (3) for a close
point s ∈ B. Since all the members of D are plane quartics thetotal space D naturally embeds in P2 × B. Moreover, without
loss of generality we can assume B to be smooth projectiveof dimension 1, or even more, we can choose B such that its
completion is contained in P1 (one could also assumed B to

7.3. Properness of the stack147

be the spectrum of a local ring in which s is the only closed
point and b is the generic one). Notice that such a deformation
always exists because we can consider the line in the Hilbert
Scheme of quartics containing the conic and two lines. The
affine open set in the line which does not contain any (too)singular quartic will give the desired deformation.
Now we claim that D ⊂ P2 × B is a normal crossing divisorin P2 × B. First we note that, up to restricting B if necessary,
we may assume that D has normal crossing singularities. Finally D can be chosen to be defined locally by one equation
depending on the uniformizing parameter of B. Such equations defines D inside P2 × B. Here we used the fact that we
can choose a generic normal crossing D (3) in any way since
Theorem 7.1.6 applies to every degree 4 and three component
plane curve, provided it has normal crossing singularities.
From the fact that D is a normal crossing divisor we have thatP2 × B has a natural fs log structure M with the property thaton each fiber P2 = P2b of the second projection the restrictionof M gives the log structure defined by the normal crossing
divisor Db . This exhibits P2 × B as a log scheme with a log
map to B. Such map has a flat underlying map as a map ofschemes. Moreover, being D → B flat, it gives a log map
P2 × B, with B with trivial log structure.
Now Theorem 6.3.10 together with Proposition 7.3.1 applied
to P2 × B gives the conclusion.
Example 7.3.4. Considering the case of two conics, and a flatfamily given by a constant irreducible conic C1 given by f = 0
and a conic C2,t degenerating to two lines as
x 2 − y2 = tover B = Spec C[t]. On P2 × B the divisor D given by x2 −y2 − t = 0 and f = 0 is a normal crossing divisor provided
that C1 ∩ C2,t intersect transversally. Up to reducing B we canalways assume this to hold. Then over P2 × B the log struc-148

7. Complements of a very generic quartic

ture given by D is fs and the restriction to every fiber P2t is thelog structure given by C1 + C2,t .

7.3.2

Properness of the moduli space of log-stable maps

The end of the proof of Theorem 7.3.2 now follows from theresults proved in the previous sections.
Proof of Theorem 7.3.2. Let D be a plane quartic and denotedby D (3) a conic and two lines in general position. By Theorem
7.2.5 we only need to show that there exists a constant A0 suchthat KΓ (P2 , M D ) is empty for all non A0 -admissible discrete
data Γ. By Proposition 7.3.3 we have that KΓ (P2 , M D ) coincide with KΓ (P2 , M D(3) ). At the same time, by Corvaja andZannier’s result, Theorem 3.2.3, and by Proposition 7.2.1, thestack KΓ (P2 , M D(3) ) is empty for all non admissible Γ withrespect to the constant A = 215 · 35. Then the same holds
for the stack with respect to the log structure coming from D
finishing the proof. In particular this gives vanishing for the
stack with non A-admissible discrete data, and hence algebraic hyperbolicity, with the same constant appearing in Theorem 7.1.6.
Remark 7.3.5. We stress that the very generic hypothesis hereis necessary since for every discrete data Γ the emptiness ofthe stack extends to the generic member of the deformation.
This is equivalent to assume that for every Γ the result holds
away from a finite number of quartics. At the same time toprove algebraic hyperbolicity we need to consider the moduli space KΓ (P2 , M D ) for infinitely many, although countably
many, discrete data. This implies that the result holds awayfrom countably many closed subset of the parameter space ofplane quartics.
The discreteness of the Γ follows from the fact thatβ ∈ H 2 (P2 , Z) ' Z

7.3. Properness of the stack149

and given g, n and β the choices of possible contact orders ~c
are finite.
In particular Theorem 7.3.2 gives a weaker result than Theorem 7.1.6 for complements of P2 in which the log irregularity
is less than 2. While it uses in an essential way the fact theCorvaja and Zannier’s result, as well as its extensions, applies
to a generic quartic with three irreducible components, our result gives genericity only for a fixed discrete data Γ but cannot
recover such genericity for proving algebraic hyperbolicity.

150

7. Complements of a very generic quartic

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