Integral points and orbits of endomorphisms on the projective plane
aa r X i v : . [ m a t h . N T ] A p r Integral points and orbits of endomorphisms onthe projective plane
Aaron Levin ∗ Yu Yasufuku † Abstract
We analyze when integral points on the complement of a finite unionof curves in P are potentially dense. We divide the analysis of these affinesurfaces based on their logarithmic Kodaira dimension ¯ κ . When ¯ κ = −∞ ,we completely characterize the potential density of integral points in termsof the number of irreducible components on the surface at infinity andthe number of multiple members in a pencil naturally associated to thesurface. When integral points are not potentially dense, we show that theylie on finitely many effectively computable curves. When ¯ κ = 0, we provethat integral points are always potentially dense. The bulk of our analysisconcerns the subtle case of ¯ κ = 1. We determine the potential density ofintegral points in a number of cases and develop tools for studying integralpoints on surfaces fibered over a curve. Finally, nondensity of integralpoints in the case ¯ κ = 2 is predicted by the Lang-Vojta conjecture, towhich we have nothing new to add.In a related direction, we study integral points in orbits under endo-morphisms of P . Assuming the Lang–Vojta conjecture, we prove that anorbit under an endomorphism φ of P can contain a Zariski-dense set ofintegral points (with respect to some nontrivial effective divisor) only ifthere is a nontrivial completely invariant proper Zariski-closed set withrespect to φ . This may be viewed as a generalization of a result of Silver-man on integral points in orbits of rational functions. We provide manyspecific examples, and end with some open problems.Mathematics Subject Classification (2010): 14J20, 14R05, 14G40, 11G35,37P55 In this paper we will study integral points on affine open subsets of P , and themore specific problem of integral points that lie in an orbit of an endomorphismof P . Our analyses are based on structure theorems for affine surfaces classifiedby their logarithmic Kodaira dimension. ∗ Supported in part by NSF grant DMS-1102563. † Supported in part by JSPS Grant-in-Aid 15K17522 and by Nihon University College ofScience and Technology Grant-in-Aid for Fundamental Science Research. C ⊂ A N over a number field k has only finitely many integral points if eitherthe curve C has positive genus or the curve C is rational and has more than twopoints at infinity. For curves of genus g ≥
2, Siegel’s theorem is superseded byFaltings’s theorem that the set of rational points C ( k ) is finite. Both theoremsmay be unified into the single statement that a curve of log general type has onlyfinitely many integral points (note that for a projective curve, integral points arethe same as rational points). In higher dimensions, the Lang–Vojta conjecturepredicts that this unified statement continues to hold, with finiteness replacedby Zariski non-density: Conjecture 1 (Lang–Vojta) . Let V be a variety defined over a number field k and let S be a finite set of places of k containing the archimedean places. If V is of log general type, then any set of S -integral points on V is not Zariski densein V . This conjecture is a consequence of a much more general height inequalityconjectured by Vojta [30, Conjecture 3.4.3]. When V is a projective surface theconjecture was formulated by Bombieri.In higher dimensions, outside of some important special cases (e.g., sub-varieties of semiabelian varieties [12, 13, 31, 33]), not much is known towardsConjecture 1. For instance, if V = P \ D , where D is a nonsingular plane curve,then V is of log general type if and only if deg D ≥
4. However, there is not asingle such V for which Conjecture 1 is known for all applicable k and S . Inview of this, in studying integral points on affine subsets of P we will oftenwork under the assumption of Conjecture 1.Suppose now that V is an affine surface given as the complement in P ofa (possibly reducible) curve D defined over a number field k . A basic invariantof V is the log Kodaira dimension ¯ κ ( V ), whose definition will be recalled inSection 2. Here we just note that ¯ κ ( V ) ∈ {−∞ , , , } and ¯ κ ( V ) = 2 if andonly if V is of log general type. If V = P \ D , where D is a normal crossingsdivisor, then ¯ κ ( V ) = −∞ , if deg D = 1 , , if deg D = 32 , if deg D ≥ . However, if D is not a normal crossings divisor, then computing ¯ κ ( V ) ismore subtle. A first goal of this paper is to understand integral points on V via the invariant ¯ κ ( V ). If ¯ κ ( V ) = 2 then, as discussed, this is accomplished(conjecturally) by Lang–Vojta’s conjecture. To avoid situations where the lackof integral points is caused by special arithmetic properties of certain numberfields or certain sets of primes, we analyze potential density of integral points,namely whether there exist a number field L ⊃ k and a finite set of places S of L for which V ( O L,S ) is Zariski-dense. 2hen ¯ κ ( V ) = −∞ , we prove the following result using structure theory foraffine surfaces: Theorem 2.
Let D be an effective divisor on P defined over Q and let V = P \ D . Suppose that ¯ κ ( V ) = −∞ . Let Λ be the associated pencil of Miyanishiand Sugie (Theorem 27). Let D ′ be the union of D and the multiple membersof Λ . Let r be the number of irreducible components of D ′ . Then integral pointson V are potentially dense if and only if r ≤ . Moreover, if r ≥ , then forany finite set of places S of k containing the archimedean places, V ( O k,S ) iscontained in the union of finitely many effectively computable curves. When ¯ κ ( V ) = 0 it turns out that integral points on V are always potentiallydense: Theorem 3.
Let D be an effective divisor on P defined over Q and let V = P \ D . Suppose that ¯ κ ( V ) = 0 . Then integral points on V are potentially dense. This agrees with a very general conjecture of Campana [8, Th. 5.1, Conj. 9.20]that if V is a projective variety and ¯ κ ( V ) = 0, then rational points on V arepotentially dense.When ¯ κ ( V ) = 1, the situation is more subtle. In this case, we know fromKawamata [21] that there is a pencil Λ of curves in P whose restriction to V yields a G m -fibration. Kojima [22] has proved a structure theorem for this case,describing a certain open subset P \ D ′ ⊂ V , but unlike the ¯ κ = −∞ case,additional components in D ′ come from singular fibers, rather than multiplefibers. This makes it harder to retrieve Diophantine information about D fromthat of D ′ .In Section 3.3, we will define two weights associated to ( D, Λ), the gcd-weight and the Campana weight. These weights measure the multiplicities of themembers of Λ, using either the gcd of the multiplicities appearing in a memberof the pencil or the minimum (following Campana [8]). Our definitions of theweights also take into account the divisor D with respect to which integrality isdefined. We prove the following result relating these weights to integral points: Theorem 4 (cf. Theorem 23) . Let D be an effective divisor on P definedover a number field k and let V = P \ D . Let S be a finite set of places of k containing the archimedean places. Let Λ be a pencil of curves on P such thatthe base points of Λ are contained in the support of D .(i) If the gcd-weight of ( D, Λ) is greater than , then the S -integral points of P \ D are contained in the union of finitely many members of Λ . Fur-thermore, if the support of D contains the support of a member of Λ , thenthis finite union is effectively computable.(ii) If the Campana weight of ( D, Λ) is greater than , then the abc -conjectureimplies that the S -integral points of P \ D are contained in the union offinitely many members of Λ . κ = 1 case,we will use both parts of Theorem 4, but mainly (ii). Among our results in thiscase, we just mention the following result here: Theorem 5 (cf. Theorems 35 and 37) . Let D be a rational curve defined over Q having just cusps as singularities, and let V = P \ D . Suppose that ¯ κ ( V ) = 1 .Then assuming the abc conjecture, integral points on V are potentially denseonly if D is projectively equivalent to one of the five types of curves listed inTheorems 35 (i) and 37. We also analyze potential density of integral points for other varieties V with ¯ κ ( V ) = 1, such as when deg D is small or when D consists of a line and anirreducible curve. Furthermore, in many cases of Theorem 5, we can actuallyconstruct Zariski-dense sets of integral points, providing a partial converse tothe theorem. We will defer the precise statements to Theorems 35–40 and theexamples in Section 7.Our proofs of Theorems 2–5 combine results from affine algebraic geometrywith results from Diophantine analysis. From geometry, key ingredients in ourproofs include the structure theory of surfaces V = P \ D with ¯ κ ( V ) = −∞ (Miyanishi-Sugie [23]) and ¯ κ ( V ) ≤ κ = 1, due to Tono [26, 27]. From Diophantine analysis, we use various resultsincluding Darmon-Granville’s results [11], unit equations, and Baker’s theory toprove Theorem 4.We now discuss integral points in an orbit under an endomorphism of P .Here we denote by φ n the n -th iterate of φ , and by O φ ( α ) the orbit { α, φ ( α ) , φ ( α ) , . . . } of α . Our starting point is Silverman’s theorem on integral points in orbits ofrational functions. Theorem 6 (Silverman [25]) . Let φ ( z ) ∈ k ( z ) be a rational function of degree d ≥ , over a number field k , with the property that φ ( z ) is not a polynomial.Let S be a finite set of places of k containing the archimedean places. Then forany α ∈ k , the orbit O φ ( α ) contains only finitely many S -integral points. If φ ( z ) ∈ k [ z ], then for an appropriate choice of S and α , O φ ( α ) will containinfinitely many S -integral points. Note also that if φ ( z ) is a polynomial, then E = {∞ , φ ( ∞ ) } is a completely invariant set for φ , that is, φ − ( E ) = E = φ ( E ).Thus Silverman’s theorem easily implies: Corollary 7.
Let φ : P → P be an endomorphism defined over a number field k . Let S be a finite set of places of k containing the archimedean places. Let D be a nontrivial effective divisor on P . If O φ ( P ) contains infinitely many S -integral points in ( P \ D )( O k,S ) for some point P , then there exists a nonemptycompletely invariant finite subset of P under φ .
4n view of Corollary 7, the second author has asked the following higher-dimensional analogue:
Question 8 (cf. [36, Question 2]) . Let φ : P n → P n be an endomorphismdefined over a number field k . Let S be a finite set of places of k containing thearchimedean places. Let D be a nontrivial effective divisor on P n . If O φ ( P ) ∩ ( P n \ D )( O k,S ) is Zariski dense in P n for some point P , then does there exist anonempty completely invariant proper Zariski-closed subset of P n under φ ?Assuming Lang–Vojta’s conjecture, we will use our analysis of integral pointson open affine subsets of P to answer this question positively. Theorem 9 (cf. Theorem 42) . Assume Lang–Vojta’s conjecture. Let φ : P → P be an endomorphism of degree > defined over a number field k . Let S be a finite set of places of k containing the archimedean places. Let D be anontrivial effective divisor on P defined over k . If O φ ( P ) ∩ ( P \ D )( O k,S ) isZariski dense in P for some point P , then there exists a nonempty completelyinvariant proper Zariski-closed subset of P under φ . Finally, we remark that our results should admit, via Vojta’s dictionary [30],corresponding results in Nevanlinna theory and for holomorphic maps f : C → P \ D . We will not pursue this here, but content ourselves with a few remarks.Recall that via Vojta’s dictionary, an infinite (or Zariski-dense) set of ( D, S )-integral points on P corresponds to a holomorphic map f : C → P \ D that isnonconstant (or with Zariski-dense image). Since our proofs combine geometricresults with arguments from Diophantine approximation involving (local) heightfunctions, all of our proofs and results on nondensity of integral points on P \ D should translate, in a straightforward manner, to corresponding proofs involvingNevanlinna theory and results on holomorphic maps f : C → P \ D . One notabledifference is that in Nevanlinna theory the analogue of the abc conjecture isknown (Nevanlinna’s Second Main Theorem with truncated counting functions).Thus, our results on integral points that are conditional on the abc conjecturewill yield unconditional corresponding results for holomorphic curves.The paper is organized as follows. In Section 2, we recall some basic defini-tions from algebraic geometry and Diophantine geometry. In Section 3, we col-lect together theoretical results on integral points, including a geometric lemma(Lemma 25) used to produce Zariski dense sets of integral points. In this section,we also define the two notions of weights of pencils, and discuss their relation tointegral points (Theorem 23). In Section 4, we quote various structure theoremsfor affine subsets of P . Section 5 is the heart of the paper and we work to-wards classifying divisors D for which P \ D has potentially dense set of integralpoints, using results of Sections 3 and 4. In Section 6, we prove a more preciseversion of Theorem 9 (Theorem 42). In Section 7, we discuss a series of specificexamples. We end with several open problems for further study.5 Definitions and Notation
In this section, we recall some definitions and notations from algebraic geometryand Diophantine geometry. See [7, 18, 19] for further details. We begin bydefining the Kodaira-Iitaka dimension κ of a divisor. Definition 10.
Let D be a divisor on a smooth projective variety X over analgebraically closed field of characteristic 0. If dim H ( X, O ( nD )) = 0 for all n > κ ( X, D ) = −∞ . Otherwise, define κ = κ ( X, D ) to be theinteger for which lim sup n →∞ dim H ( X, O ( nD )) n κ exists and is nonzero.It is well-known that κ ( X, D ) ∈ {−∞ , , , . . . , dim X } . We now define thelogarithmic Kodaira dimension of a quasi-projective variety. Definition 11.
Let V be a smooth quasi-projective variety over an algebraicallyclosed field of characteristic 0. Let V = X \ D , where X is a smooth projectivevariety and D is a normal crossings divisor on X . Then the log Kodaira di-mension ¯ κ ( V ) of V is defined to be the Kodaira-Iitaka dimension of the divisor K X + D , where K X is the canonical divisor of X .Note that there always exists a choice of X and D as in the definition (byHironaka’s results) and it is known that κ ( K X + D ) is independent of the choiceof X and D (so that ¯ κ ( V ) is well-defined). We say that V is of log general type if ¯ κ ( V ) is as large as possible, that is, if ¯ κ ( V ) = dim V .Let k be a number field, and let us now assume that the smooth projectivevariety X and the effective divisor D are both defined over k . There are severalessentially equivalent ways of defining a set of integral points on X \ D , includingthe natural scheme-theoretic definition coming from a choice of integral modelof X \ D . Here we will follow Vojta [30] and define sets of integral points vialocal height functions. Let M k be the canonical set of places of k , consistingof one place for each prime ideal p of O k , one place for each real embedding σ : k → R , and one place for each pair of conjugate embeddings σ, σ : k → C .For each v ∈ M k , let | · | v denote the corresponding absolute value, normalizedso that | p | v = p − [ k v : Q v ] / [ k : Q ] if v corresponds to a prime p and p lies above arational prime p , and | x | v = | σ ( x ) | [ k v : Q v ] / [ k : Q ] if v corresponds to an embedding σ . With this normalization, the product formula Y v ∈ M k | x | v = 1holds for all x ∈ k ∗ . Let Supp D denote the support of D . From the theoryof heights, for each place v ∈ M k we can associate to D a local height function λ v ( D, − ) : X ( k ) \ Supp D → R , unique up to a bounded function, such that X v ∈ M k λ v ( D, P ) = h ( D, P ) + O (1)6or all P ∈ X ( k ) \ Supp D , where h ( D, − ) is a Weil height with respect to D .We will not give a precise definition here, but λ v ( D, P ) is, up to a boundedfunction, − log | f ( P ) | v , where f is a local equation of D around P . We recallthat both global and local heights are functorial with respect to pullbacks bymorphisms, in the following sense: if φ : Y → X is a morphism of smoothprojective varieties with φ ( Y ) Supp D , then λ v ( φ ∗ D, P ) = λ v ( D, φ ( P )) + O (1) ,h ( φ ∗ D, P ) = h ( D, φ ( P )) + O (1) . In fact, since the global (Weil) height depends only on the linear equivalenceclass of the divisor D , if one works with divisor classes (or line bundles) thecondition φ ( Y ) Supp D may be avoided for the global height.Now, let S be a finite subset of M k containing all the archimedean places.Then a set of ( k -rational) ( D, S )- integral points on X is defined to be a set ofthe form P ∈ X ( k ) \ Supp D : X v ∈ M k \ S λ v ( D, P ) ≤ C for some choice of local height functions and for some constant C . This notiondepends only on the support of D . In fact, it depends only on the variety V = X \ D . Thus, for a smooth quasi-projective variety V over k we willcall a subset R ⊂ V ( k ) a set of S -integral points on V if there exist a smoothprojective variety X and an effective divisor D on X such that V = X \ D and R is a set of ( D, S )-integral points on X . We say that integral points on V are potentially dense if there exists a Zariski-dense set of S -integral points on V forsome number field k and some finite subset S ⊂ M k .On P n , one can choose the local height functions in a canonical way, and thusdefine sets of integral points unambiguously. For a polynomial f ∈ k [ x , . . . , x n ]and v ∈ M k , we let | f | v denote the maximum of the v -adic absolute values of thecoefficients of f . Now let D be a hypersurface in P n defined by a homogeneouspolynomial f ∈ k [ x , . . . , x n ] of degree d . For v ∈ M k and P = ( x , . . . , x n ) ∈ P n ( k ) \ Supp D , x , . . . , x n ∈ k , we define the local height function λ v ( D, P ) = log | f | v max i | x i | dv | f ( P ) | v . (1)This definition is independent of the choice of the defining polynomial f andthe choice of the coordinates for P . Note also that for a nonarchimedean v , λ v ( D, P ) ≥ P ∈ P n ( k ) \ Supp D . Then we can define the set of S -integral points ( P n \ D )( O k,S ) to be the set of points P ∈ P n ( k ) \ Supp D : X v ∈ M k \ S λ v ( D, P ) = 0 . v ∈ M k , we define the minimum positive valuation ν v to be ν v = min {− log | x | v : x ∈ k ∗ , − log | x | v > } . (2)We can then define the truncated local height for a non-archimedean v by λ (1) v ( D, P ) = min ( λ v ( D, P ) , ν v ) . This captures the nontrivial contribution to the v -adic local height, but onlycounted with the smallest possible contribution. In this section, we collect together theoretical results on integral points that wewill use. These results will be combined with the geometric structure theorems ofthe next section to yield a study of integral points on affine subsets of P . Someof our results are also conditional on well-known conjectures, whose statementswe now recall. We discuss several variants of the abc conjecture. † The following is the moststandard formulation:
Conjecture 12 (Masser–Oesterl´e abc conjecture) . For all ǫ > , there is aconstant C > such that for all a, b, c ∈ Z with a + b + c = 0 and gcd( a, b, c ) = 1 ,we have max {| a | , | b | , | c |} ≤ C Y p | abc p ǫ . Translating the conjecture into the language of heights (see [32] for the de-tails) and allowing for arbitrary number fields and (finite) sets of places S , weobtain the following formulation of the abc conjecture for number fields. Conjecture 13 ( abc conjecture for number fields) . Let k be a number field andlet S be a finite set of places of k containing the archimedean places. Let ǫ > .There exists a constant C such that for all x ∈ P ( k ) \ { , , ∞} , (1 − ǫ ) h ( x ) ≤ X v ∈ M k \ S (cid:16) λ (1) v ((0) , x ) + λ (1) v ((1) , x ) + λ (1) v (( ∞ ) , x ) (cid:17) + C. † Mochizuki [24] has claimed a proof of this conjecture. k = Q and S = ∞ is precisely the conjecture of Masserand Oesterl´e. Using Belyi maps, Conjecture 13 is equivalent to the followingconjecture, which replaces the three points { , , ∞} by an arbitrary finite setof points on the projective line (see [29] for a precise relationship between thetwo conjectures). Conjecture 14 (general abc conjecture) . Let k be a number field and let S bea finite set of places of k containing the archimedean places. Let α , . . . , α q ∈ P ( k ) be distinct points and let ǫ > . There exists a constant C such that forall x ∈ P ( k ) \ { α , . . . , α q } , ( q − − ǫ ) h ( x ) ≤ q X ℓ =1 X v ∈ M k \ S λ (1) v (( α ℓ ) , x ) + C. (3)It is this form of the abc conjecture that will be the most convenient for ourapplications. We note that Conjecture 14 is the special case for P of a generalconjecture of Vojta valid for all smooth complete varieties [32, Conjecture 2.3]. In this section, we define two weights associated to a fibration over P , and dis-cuss their importance relative to integral points. In a general setting, Campana[8] associates to a fibration f : V → W a certain Q -divisor on W reflecting themultiple fibers of the fibration, and uses this divisor to give the base of the fibra-tion the structure of an orbifold (in the sense of Campana), called the orbifoldbase of the fibration. One can then define the canonical bundle and Kodairadimension of this orbifold base. As in the Lang-Vojta conjecture (Conjecture1), when the Kodaira dimension of the orbifold base is maximal, one expectsintegral points on the orbifold base to be sparse, and then the same conclusionholds for the variety V . We now make this more precise in the special caseof a fibration over P , where all of the relevant information is captured in thedefinition of certain weights associated to the fibration. We first define a weightusing the minimum of the multiplicities in a fiber, following Campana [8]. Definition 15.
Let D be an effective divisor on a nonsingular projective variety V over a field k of characteristic 0. Let φ : V → P be a nonconstant morphismover k . For each point P ∈ P (¯ k ), we define the Campana multiplicity m D,φ ( P )to be the infimum of the multiplicities of the irreducible components of the fiberover P , excluding any irreducible components of D . The infimum of the emptyset is defined as usual to be ∞ , so if all components of a fiber φ − ( P ) are inside D , then m D,φ ( P ) is defined to be ∞ . Then the Campana weight of (
D, φ ) is X P ∈ P (¯ k ) (cid:18) − m D,φ ( P ) (cid:19) . Remark 16.
Campana dealt with the case D = 0 in the above definition, wherethe infimum is taken over all irreducible components of the fibers. If we use the9onvention that every component of D has multiplicity ∞ in any fiber, thenwe can still define m D,φ ( P ) to be the infimum of the multiplicities of all theirreducible components of the fiber over P , including those inside D .The significance of the Campana weight of ( D, φ ) is contained in the nextresult, conditional on the abc conjecture.
Theorem 17.
Assume the abc conjecture (Conjecture 14). Let D be an effectivedivisor on a nonsingular projective variety V , both defined over a number field k . Let φ : V → P be a nonconstant morphism over k . Let S be a finite set ofplaces of k containing the archimedean places. If the Campana weight of ( D, φ ) is greater than , then any set of S -integral points on V \ D is contained infinitely many fibers of φ . In particular, any set of S -integral points on V \ D isZariski-non-dense.Proof. Let α , . . . , α q be the points of P (¯ k ) over which φ has multiple fibers orcontains an irreducible component of D . Let φ ∗ (( α ℓ )) = X i m ℓi D ℓi + X j n ℓj F ℓj be the decomposition into irreducible components, where the D ℓi ’s are the com-ponents contained in D and the F ℓj ’s are the remaining components.Now let R be a ( D, S )-integral set of points. For P ∈ R , we derive the fol-lowing from Conjecture 14 and functoriality of heights with respect to pullbacksby morphisms:( q − − ǫ ) h ( φ ( P )) ≤ q X ℓ =1 X v ∈ M k \ S λ (1) v (( α ℓ ) , φ ( P )) + O (1) ( ∵ (3))= q X ℓ =1 X v ∈ M k \ S λ (1) v X i m ℓi D ℓi + X j n ℓj F ℓj , P + O (1)= q X ℓ =1 X v ∈ M k \ S λ (1) v X j n ℓj F ℓj , P + O (1) ( ∵ P S -integral)= q X ℓ =1 X v ∈ M k \ S λ (1) v X j F ℓj , P + O (1) ≤ q X ℓ =1 h X j F ℓj , P + O (1) ≤ q X ℓ =1 j n ℓj h X j n ℓj F ℓj , P + O (1) (4) ≤ q X ℓ =1 j n ℓj h X i m ℓi D ℓi + X j n ℓj F ℓj , P + O (1)10 q X ℓ =1 j n ℓj h (( α ℓ ) , φ ( P )) + O (1)= q X ℓ =1 m D,φ ( α ℓ ) ! h ( φ ( P )) + O (1) . Note that the inequality (4) holds even when { n ℓj } j is the empty set for some ℓ . Since the Campana weight of ( D, φ ) is assumed to be greater than 2, P qℓ =1 1 m D,φ ( α ℓ ) < q −
2, and by choosing ǫ sufficiently small, we conclude that h ( φ ( P )) is bounded for P ∈ R . Hence, P must be contained in the union offinitely many fibers of φ .To obtain unconditional (and even effective) results, we consider anotherversion of multiplicities and weights, replacing inf by gcd. This construction isin fact more classical in algebraic geometry. Definition 18.
Let D be an effective divisor on a nonsingular projective variety V over a field k of characteristic 0. Let φ : V → P be a nonconstant morphismover k . For each point P ∈ P (¯ k ), we define the classical multiplicity m − D,φ ( P )to be the greatest common divisor of the multiplicities of the irreducible com-ponents of the fiber over P , excluding any irreducible components of D . Byconvention, we define the gcd of the empty set to be ∞ , so if all componentsof a fiber φ − ( P ) are inside D , then m − D,φ ( P ) is defined to be ∞ . Then the gcd-weight of ( D, φ ) is X P ∈ P (¯ k ) − m − D,φ ( P ) ! . One can give an unconditional version of Theorem 17 using the gcd-weightof (
D, φ ). This is essentially due to Darmon and Granville [11] (see also [10]).
Theorem 19.
Let D be an effective divisor on a nonsingular projective variety V , both defined over a number field k . Let φ : V → P be a nonconstantmorphism over k . Let S be a finite set of places of k containing the archimedeanplaces. If the gcd-weight of ( D, φ ) is greater than , then any set of S -integralpoints on V \ D is contained in finitely many fibers of φ . In particular, any setof S -integral points on V \ D is Zariski-non-dense.Proof. Let R be a set of S -integral points on V \ D and let P ∈ R . Let { P , . . . , P n } = { P ∈ P (¯ k ) | m − D,φ ( P ) > } and replace k by the finite extension k ( P , . . . , P n ). Set m i = m − D,φ ( P i ). Thenfor each i , we have Supp φ ∗ P i ⊂ Supp D if m i = ∞ , and otherwise φ ∗ P i = D i + m i F i for some effective divisors D i and F i on V with Supp D i ⊂ Supp D .It follows from functoriality of local heights that λ v ( P i , φ ( P )) = m i λ v ( F i , P ) + O v (1) , m i < ∞ , (5)11 v ( P i , φ ( P )) = O v (1) , m i = ∞ , (6)for all v ∈ M k \ S , where O v (1) = 0 for all but finitely many v . Since λ v ( P i , φ ( P ))can be taken as the product of a v -adic intersection pairing ( P i , φ ( P )) v and ν v of(2), it follows that for some finite set of places S ′ of k containing the archimedeanplaces, we can choose an integral model on P so that the set φ ( R ) is containedin a set of S ′ -integral points on the M -curve ( P ; P , m ; · · · ; P n , m n ), in thelanguage of Darmon [10]. Our assumption on the gcd-weight of ( D, φ ) impliesthat this M -curve has negative Euler characteristic, and so by [10, Theorem(Faltings plus epsilon)], φ ( R ) is a finite set. Thus, R lies in finitely many fibersof φ .Under the additional assumption that one of the multiplicities is infinite,one obtains an effective result. This is essentially due to Bilu [6, Th. 1.2], whoproved an explicit quantitative result when V is a curve. Theorem 20.
Let D be an effective divisor on a nonsingular projective variety V , both defined over a number field k . Let φ : V → P be a nonconstantmorphism over k . Let S be a finite set of places of k containing the archimedeanplaces. If the gcd-weight of ( D, φ ) is greater than and m − D,φ ( P ) = ∞ for somepoint P ∈ P (¯ k ) , then any set of S -integral points on V \ D is contained infinitely many effectively computable fibers of φ . Implicit in Theorem 20, we assume that a set of S -integral points on V \ D is given in an explicit fashion such that the constants in (5) and (6) in the proofof Theorem 19 are effectively computable. Proof.
As before, let R be a set of S -integral points on V \ D and let { P , . . . , P n } = { P ∈ P (¯ k ) | m − D,φ ( P ) > } . Replacing k by the finite extension k ( P , . . . , P n ), we may assume that P , . . . , P n are k -rational. We consider four cases.Case I: There are at least three distinct points P i with m − D,φ ( P i ) = ∞ . Afteran automorphism of P , we can assume that m − D,φ (0) = m − D,φ (1) = m − D,φ ( ∞ ) = ∞ . Then we can enlarge S so that φ ( P ) and 1 − φ ( P ) are S -units for all P ∈ R .Thus, setting u = φ ( P ) and v = 1 − φ ( P ), we obtain a solution to the S -unitequation u + v = 1 , u, v ∈ O ∗ k,S . As the S -unit equation has finitely many solutions, which are effectively com-putable, it follows that φ ( R ) is a finite set and R is contained in finitely manyeffectively computable fibers of φ .Case II: There are exactly two distinct points P i with m − D,φ ( P i ) = ∞ . Afteran automorphism of P , we can assume that m − D,φ (0) = m − D,φ ( ∞ ) = ∞ . Sincethe gcd-weight of ( D, φ ) is greater than 2, there is at least one other point P i m − D,φ ( P i ) = m , 1 < m < ∞ . After an automorphism of P , we may assumethat m − D,φ (1) = m . Then we can enlarge S so that φ ( P ) is an S -unit and theideal generated by φ ( P ) − m th power of an ideal in O k,S for all P ∈ R .In fact, after enlarging S so that O k,S is a principal ideal domain and adjoiningthe m th roots of the finitely many generators of O ∗ k,S , we obtain a number field L and a finite set of places T of L such that φ ( P ) is a T -unit and φ ( P ) − m th power in O L,T for all P ∈ R . Let u = φ ( P ). Then u satisfies the equation u − z m , u ∈ O ∗ L,T , z ∈ O
L,T . (7)Rewriting this as u = z m + 1 and noting that the right hand side has at leasttwo distinct roots (over ¯ k ), the equation (7) easily reduces to unit equations. Itfollows that φ ( R ) is a finite set and R is contained in finitely many effectivelycomputable fibers of φ .Case III: There is exactly one point P i with m − D,φ ( P i ) = ∞ and there isat least one point P j with finite multiplicity m − D,φ ( P j ) >
2. Note that by ourassumptions there must exist a third point P k , distinct from P i and P j , withfinite multiplicity m − D,φ ( P k ) >
1. After an automorphism of P , we can assumethat m − D,φ ( ∞ ) = ∞ ,m − D,φ (0) = m ≥ ,m − D,φ (1) = n ≥ . Then arguing as before, there exist a number field L and a finite set of places T of L , such that φ ( P ) = x m for some x ∈ O L,T and x m − y n for some y ∈ O L,T for all P ∈ R . Since m, n > m, n ) = (2 , x m − y n = 1 , x, y ∈ O L,T has only finitely many effectively computable solutions. It follows that φ ( R ) isa finite set and R is contained in finitely many effectively computable fibers of φ .Case IV: There is exactly one point P i with m − D,φ ( P i ) = ∞ and there are atleast three points P j with multiplicity m − D,φ ( P j ) = 2. After an automorphism of P , we can assume that m − D,φ ( ∞ ) = ∞ and m − D,φ (0) = m − D,φ ( a ) = m − D,φ ( b ) = 2for some distinct a, b ∈ k ∗ . Then arguing as before, there exist a number field L and a finite set of places T of L such that for all P ∈ R , φ ( P ) = x for some x ∈ O L,T and x − a = y , (8) x − b = z , (9)for some y, z ∈ O L,T . As is well-known, the equations (8) and (9) yield an affinemodel of an elliptic curve. Since Siegel’s theorem is effective for elliptic curves,13e again obtain that φ ( R ) is a finite set and R is contained in finitely manyeffectively computable fibers of φ .It’s clear that every possible case is covered by Cases I-IV, finishing theproof. In this section we reformulate the definitions and results of the last section inthe context of pencils of plane curves.
Definition 21.
Let Λ be a pencil of curves on P and let D be an effectivedivisor on P containing the base points of Λ. For each member C ∈ Λ, let m D, Λ ( C ) (resp. m − ( D, Λ)( C )) be the infimum (resp. gcd) of the multiplicitiesof the irreducible components of C , excluding irreducible components which arealso components of D . Then the Campana weight of ( D, Λ) is X C ∈ Λ (cid:18) − m D, Λ ( C ) (cid:19) and the gcd weight of ( D, Λ) is X C ∈ Λ − m − D, Λ ( C ) ! . We connect these definitions with the results of the last section through thefollowing lemma.
Lemma 22.
Let Λ be a pencil of curves on P and let D be an effective divisor on P containing the base points of Λ . Let φ = φ Λ : P P be the correspondingrational map. Let ˜ φ : V → P be a morphism resolving the indeterminacy locusof φ , that is a map for which V ˜ φ ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ π (cid:15) (cid:15) P φ / / ❴❴❴❴❴❴ P commutes where π : V → P is a birational morphism of nonsingular varieties.The Campana weight of ( D, Λ) is the same as the Campana weight of ( π ∗ D, ˜ φ ) .The gcd weight of ( D, Λ) is the same as the gcd weight of ( π ∗ D, ˜ φ ) .Proof. Let P ∈ P (¯ k ). Let C P ∈ Λ be the corresponding member of the pencil.Since D contains the base points of Λ, when computing m π ∗ D, ˜ φ ( P ) we ignoreany exceptional divisors of V lying over base points of Λ. It’s clear then that m π ∗ D, ˜ φ ( P ) = m D, Λ ( C P ) and m − π ∗ D, ˜ φ ( P ) = m − D, Λ ( C P )14sing Lemma 22 and the results of the last section, we immediately obtainthe following result which will be used repeatedly to analyze integral points onaffine subsets of P . Theorem 23.
Let Λ be a pencil of curves on P and let D be an effective divisoron P containing the base points of Λ , with D and Λ both defined over a numberfield k . Let S be a finite set of places of k containing the archimedean places.(a). Assume the abc conjecture. If the Campana weight of ( D, Λ) is greaterthan , then the S -integral points of P \ D are contained in the union offinitely many members of Λ .(b). If the gcd-weight of ( D, Λ) is greater than , then the S -integral points of P \ D are contained in the union of finitely many members of Λ .(c). If the gcd-weight of ( D, Λ) is greater than and the support of D containsthe support of a member of Λ , then the S -integral points of P \ D arecontained in the union of finitely many effectively computable members of Λ . Remark 24.
We note that Theorem 23 holds regardless of the log Kodairadimension of P \ D . Further, we note that the pencil does not necessarily haveto be the one coming from the structure theory of Kawamata [21] and Gurjar–Miyanishi [15]. That is, as long as a pencil satisfies the hypotheses, it does nothave to be the one created from the so-called ‘peeling’ theory. The following general lemma will be used several times to construct Zariski-dense sets of integral points.
Lemma 25.
Let D be an effective divisor on P defined over Q . Let Λ be apencil of plane curves with the property that for a general member C ∈ Λ , either C \ D ∼ = A or C \ D ∼ = G m (over Q ). Suppose that there is a plane curve C defined over Q such that C is not a component of any member of Λ and either C \ D ∼ = A or C \ D ∼ = G m (over Q ). Then integral points on P \ D arepotentially dense.Proof. Let k be a number field with the following properties: • The divisor D is defined over k . • Generators for the pencil Λ are defined over k . • C \ D is isomorphic over k to either A or G m . • k has at least one complex archimedean place.15ince D is very ample, we may embed P \ D over k into some affine space A N .By our assumption on C , for some finite set of places S of k containing thearchimedean places, ( C \ D )( O k,S ) = ( C \ D )( k ) ∩ A N ( O k,S ) is infinite. Wemay further assume that | S | ≥
2. For every point P ∈ C , let C P denote amember of Λ containing P . For all but finitely many P ∈ C ( O k,S ), we have C P \ D ∼ = A or C P \ D ∼ = G m (over k ). Let P be such a point and assumefurther that P is not a base point of Λ. We first note that C P is defined over k . Indeed, since Λ has generators defined over k and C P contains the k -rationalpoint P , any conjugate of C P over k would also be a member of Λ containing P ,and since P is not a base point, must coincide with C P . Let C ′ P = C P \ D . Since P ∈ C ′ P ( O k,S ) is a nonsingular point of C ′ P , C ′ P is rational and has at most twopoints at infinity, k has at least one complex archimedean place, and | S | ≥ C ′ P ( O k,S ) = C ′ P ( k ) ∩ A N ( O k,S ) is infinite.Since C is not a component of any member of Λ, C P varies as P varies, and itis clear that there exists a Zariski-dense set of ( D, S )-integral points on P .Finally, we recall a result of Beukers [5] on integral points on the complementof a nonsingular plane cubic. Theorem 26 (Beukers) . Let C be a nonsingular projective cubic plane curvedefined over a number field k . Then integral points on P \ C are potentiallydense. More precisely, Beukers [5, Th. 3.3] proves that S -integral points on P \ C are Zariski dense if C has a k -rational flex F and the tangent line to C through F is not a component of C modulo any prime outside S . Since this conditionis clearly satisfied for large enough k and S , we obtain the potential density ofTheorem 26. P P via the log-arithmic Kodaira dimension. Since we will use Lang-Vojta’s conjecture (Conjec-ture 1) to handle surfaces of log general type (¯ κ = 2), we will only be interestedin the remaining three possibilities for a surface: ¯ κ = −∞ , ,
1. Throughout thissection, we work over the complex numbers (or more generally, an algebraicallyclosed field of characteristic 0). ¯ κ = −∞ The following structure theorem of Miyanishi and Sugie [23, Theorem] (see alsoKojima [22, Theorem 1.1 (i)]) will be the key ingredient in analyzing integralpoints in this case.
Theorem 27 (Miyanishi–Sugie, Kojima) . Let V = P \ D , where D is a reducedeffective divisor on P . Suppose that ¯ κ ( V ) = −∞ . Then there exists a pencil Λ on P such that: a). Every member of Λ is irreducible.(b). The pencil Λ has at most two multiple members. If it has two distinctmultiple members aF and bG , then gcd( a, b ) = 1 .(c). The divisor D is a union of irreducible components of members of Λ .(d). The pencil Λ has a unique base point P , and for a general member C ∈ Λ , C \ { P } ∼ = A .(e). Let D ′ be the union of D and the multiple members of Λ . Let r be thenumber of irreducible components of D ′ . Then P \ D ′ ∼ = P \ { r lines through a single point P } . ¯ κ = 0 A structure theorem of Kojima is the key ingredient in this case.
Theorem 28 (Kojima [22, Theorem 1.1 (ii)]) . Let V = P \ D , where D isa reduced effective divisor on P . If ¯ κ ( V ) = 0 , then either D is a nonsingularcubic curve or V contains an open subset isomorphic to G m × G m ∼ = P \{ lines in general position } . ¯ κ = 1 In this case, while we know from Kawamata [21] and Gurjar–Miyanishi [15] thatthere is a G m -fibration over P , there is not a complete classification of the affinesurfaces V ⊂ P with ¯ κ ( V ) = 1 that is sufficient for our purposes. Instead, wegive a variety of classification results under various hypotheses. In each suchcase, in the next section we will prove results on integral points on the classifiedaffine surfaces.We begin by recalling a result of Wakabayashi which greatly restricts thepossibilities of an irreducible plane curve C with ¯ κ ( P \ C ) < Theorem 29 (Wakabayashi [34]) . Let C be an irreducible curve in P andsuppose that ¯ κ ( P \ C ) < . Then one of the following is true:(a). C is a nonsingular cubic curve.(b). C is a rational curve with at most two singular points. If C has twosingular points, then both singularities are cuspidal.In particular, if ¯ κ ( P \ C ) = 1 , then C is a rational curve with exactly one ortwo singular points. In the latter case, both singularities are cuspidal. It is elementary that in case (a), ¯ κ ( P \ C ) = 0, and if C is a nonsingularrational curve, then ¯ κ ( P \ C ) = −∞ . Tsunoda [28] showed that if C has twocuspidal singularities then ¯ κ ( P \ C ) = 1 or 2 and that ¯ κ ( P \ C ) = 0 if C is arational curve with a single cuspidal point.We now describe in more detail classifications of V = P \ D with ¯ κ ( V ) = 1in the following cases: 17a). D is a rational curve with two cusps and no other singularities.(b). D is a rational curve with a single cusp and no other singularities.(c). D is an irreducible curve with deg D ≤ D is a union of a line and an irreducible curve. Remark 30.
From Theorem 29, we see that the cases not treated by (a)–(d) above are when D is irreducible of degree at least 6 having exactly onesingularity which is not a cusp, or when D is reducible and either the numberof components is at least 3 or none of the components is a line. In this case, Tono [26, Theorem 4.1.2] has a complete classification up to pro-jective equivalence.
Theorem 31 (Tono) . Let D be a rational bicuspidal curve such that ¯ κ ( P \ D ) =1 . Given −→ v = ( v , . . . , v n +1 ) ∈ C n +1 , denote by J −→ v ( x, y, z ) the polynomial x n z + P n +1 j =1 v j x n +1 − j y j . Then there exists a sequence D , . . . , D s of curvessatisfying: (i) D is projectively equivalent to D . (ii) D s is defined by F + F = 0 , where F and F are one of the followingthree possibilities: F = y α , F = x α − α ( z + ay ) α (10) F = ( J −→ a ( x, y, z )) α , F = x ( n +1) α − α y α , α < ( n + 1) α (11) F = y α , F = x α − ( n +1) α ( J −→ a ( x, y, z )) α , ( n + 1) α < α (12) where < α < α with gcd( α , α ) = 1 , a ∈ C , and −→ a ∈ C n × C ∗ with n ≥ . (iii) For each i = 1 , . . . , s , D i − = τ − −→ a i ( D i ) , where τ −→ a i is a De Jonqui`erestransformation ( x, y, z ) ( x m i +1 , J −→ a i ( x, y, z ) , x m i y ) for −→ a i ∈ C m i × C ∗ with m i ≥ .Conversely, any curve D satisfying the above conditions defines a rational bi-cuspidal curve with ¯ κ ( P \ D ) = 1 . Here we quote a different result of Tono. In [27, Theorem 2], Tono classifies,into three different cases, unicuspidal plane curves whose complements have logKodaira dimension 1.
Theorem 32 (Tono) . Let C be a rational unicuspidal curve with ¯ κ ( P \ C ) = 1 .Then one of the following holds. There exist n, s ≥ and a , . . . , a s ∈ C with a s = 0 such that C isprojectively equivalent to the curve f s − y + s X i =2 a i f s − i x ( n +1) i − n ! µ A − f µ G ! . x n = 0 , where f = x n z + y n +1 , µ A = n + 1 , and µ G = ( n + 1)( s −
1) + 1 . (ii) There exists n ≥ such that the curve C is projectively equivalent to thecurve (cid:0) ( g n y + x n +1 ) µ A − ( g n z + 2 x n yg n + x n +1 ) µ G (cid:1) / g n = 0 , where g = xz − y , µ A = 4 n + 1 , and µ G = 2 n + 1 . (iii) There exist a positive integer n ≥ , a positive integer s , and a , . . . , a s ∈ C with a s = 0 such that C is projectively equivalent to the curve h s − ( g n y + x n +1 ) + s X i =1 a i h s − i ) g mi − n ! µ A − h µ G ! . g n = 0 , where m = µ A = 4 n + 1 , g = xz − y , h = g n z + 2 x n yg n + x m , and µ G = 2((4 n + 1) s − n ) .Conversely, any curve C defined in (i) – (iii) is a rational unicuspidal curve sat-isfying ¯ κ ( P \ C ) = 1 . ≤ with a uniquesingularity From Wakabayashi’s theorem (Theorem 29), when D is irreducible and ¯ κ = 1, D has to be a rational curve having two cusps or having one singular point. Theformer case was already treated in Section 4.3.1. The latter case with deg D ≤ Theorem 33.
Let C be a rational plane curve with a unique singularity, deg C ≤ , and ¯ κ ( P \ C ) = 1 . Let e be the multiplicity at the singular point and let N be the number of points in the normalization above the singular point. Then ( e, N ) ∈ { (3 , , (4 , , (4 , } . Suppose now that D is the union of a line and an irreducible curve. After anautomorphism of P , we may assume that the line is given by Z = 0. Let f ( x, y ) be the (dehomogenized) irreducible polynomial defining the componentof D that is not the line ( Z = 0). In this case, Aoki [4, Theorem 3.7, Lemmas3.8–3.12] has determined f ( x, y ) for which ¯ κ ( A \ ( f ( x, y ) = 0)) = 1, up tochanges of coordinates in x, y . 19 heorem 34 (Aoki) . Let D be the union of the line L defined by Z = 0 andan irreducible curve C defined by a homogeneous polynomial ˜ f ( X, Y, Z ) . Let f ( x, y ) = ˜ f ( x, y, . Suppose that ¯ κ ( P \ D ) = 1 . Then after a suitable changeof coordinates on A = P \ L , f ( x, y ) is one of the following: (i) f ( x, y ) = x a y b + 1 , where gcd( a, b ) = 1 and a, b > . (ii) f ( x, y ) = x a ( x l y + p ( x )) b + 1 , where a > , b > , l > , gcd( a, b ) =1 , deg p ( x ) < l , and p (0) = 0 . (iii) f ( x, y ) = a ( x ) y + a ( x ) , where a ( x ) and a ( x ) have no common factors, deg a < deg a , and a ( x ) has at least two distinct roots over C . (iv) f ( x, y ) = x a − y b , where a, b > and gcd( a, b ) = 1 .Conversely, the complement in A of any of the curves defined in (i) – (iv) abovesatisfies ¯ κ = 1 . P ¯ κ = −∞ Using Theorem 23 and Theorem 27, we completely classify integral points inthe case ¯ κ ( P \ D ) = −∞ , proving Theorem 2 from the introduction. Proof of Theorem 2.
We may assume that the divisor D is reduced. Let r and D ′ be as in Theorem 27. When r ≤
2, by Theorem 27, P \ D ′ is isomorphicto P \ { one line } ∼ = A or P \ { two lines } ∼ = G m × A . In both cases, itis evident that the ( D ′ , S )-integral points are potentially dense, hence so are( D, S )-integral points.Now suppose that r ≥
3. We claim that the gcd weight of ( D, Λ) is greaterthan 2. Since every member of Λ is irreducible, we have the following threepossibilities:(a). D has one irreducible component and there are two distinct multiple mem-bers aF and bG of Λ, with F and G distinct from D .(b). D has two irreducible components and there is at least one multiple mem-ber aF of Λ such that F Supp D .(c). D has at least three irreducible components.Recall that in the first case, gcd( a, b ) = 1 by Theorem 27(b).We note that given a defining homogeneous polynomial f for D (over somenumber field k ), one can effectively determine the existence of a pencil Λ sat-isfying Theorem 27 and condition (a) or (b) above. First, by factoring f (over20 k ), we can determine the number of irreducible components of D . If D has oneirreducible component and (a) holds, then D must be a member of Λ. If a, b > d = deg D , then we look at the equation f ( x, y, z ) = X i,j,ki + j + k = da a i,j,k x i y j z k a + X i,j,ki + j + k = db b i,j,k x i y j z k b in the indeterminates a i,j,k and b i,j,k . Equating monomial coefficients yields asystem of equations in a i,j,k and b i,j,k and we can (in principle) determine if thesystem of equations has a solution in ¯ k (e.g., using Gr¨obner bases). Runningover all possible integers a and b , we can thus determine if there exist a pencilΛ and F and G as in (a) (and if they exist, defining polynomials). If D hastwo irreducible components and (b) holds, then an irreducible component of D of maximal degree must be a member of Λ (and some multiple of the othercomponent must also be a member of Λ, yielding generators for Λ). By thesame argument as for case (a), we can effectively determine the existence of amultiple member aF ∈ Λ with F Supp D (and an equation for F if it exists).Thus, the geometric objects required in applying Theorem 23(c) are effectivelycomputable.Now we compute that the gcd weight of ( D, Λ) is at least1 + (1 − /
2) + (1 − / > − / > > D, Λ) is greater than 2. Since D is a union of irreducible components of members of Λ, by Theorem 23(c),( P \ D )( O k,S ) is contained in the union of finitely many effectively computablecurves (in fact, the union of finitely many effectively computable members ofΛ). ¯ κ = 0 We now prove Theorem 3, showing potential density of integral points when¯ κ ( P \ D ) = 0. Proof of Theorem 3.
By Theorem 28, D is a nonsingular cubic curve or P \ D contains an open subset isomorphic to G m × G m . If D is a nonsingular cubic,then the result follows from Theorem 26. Otherwise, potential density of ( D, S )-integral points follows from potential density of integral points on G m × G m .21 .3 ¯ κ = 1 In this section, we analyze the arithmetic of the surfaces that were classifiedin Section 4.3. As mentioned previously, this does not cover all the possiblesurfaces V ⊂ P of interest with ¯ κ ( V ) = 1. In contrast to previous sections, weare also only able to give a partial analysis of the arithmetic of the surfaces inSection 4.3. We give several examples in Section 7 which go beyond the resultsof this section.The general strategy of our analysis is as follows. Using the explicit equationsof Section 4.3, we construct a corresponding pencil Λ of curves on P . Most ofthe time, the Campana weight of ( D, Λ) is greater than 2, and so Theorem23 gives us Zariski-non-density of integral points. For the rest, we attempt toexplicitly construct a Zariski-dense set of integral points. Since there is always a G m -fibration to P for such surfaces, this amounts to constructing a horizontalcurve C as in Lemma 25. We begin by analyzing integral points on the complement of a rational bicuspidalcurve.
Theorem 35.
Suppose that D is a rational curve defined over Q having exactlytwo cusps and no other singularities and that V = P \ D satisfies ¯ κ ( V ) = 1 .(i) Assume the abc conjecture. If integral points on V are potentially densethen D is projectively equivalent to one of the following: X n +1 Y + n +1 X j =1 b j X n +1 − j ) ( XZ + cY ) j + X ( XZ + cY ) n +1 = 0 Y α +1 + X ( Z + aY ) α = 0 XY ( n +1) α − + X n Z + n +1 X j =1 a j X n +1 − j Y j α = 0 Y ( n +1) α +1 + X X n Z + n +1 X j =1 a j X n +1 − j Y j α = 0 where n ≥ , α ≥ , a, a j , b j ∈ Q , and a n +1 , b n +1 , c ∈ ( Q ) ∗ .(ii) Suppose that D is defined by one of the following: Y α +1 + X ( Z + aY ) α = 0 XY ( n +1) α − + (cid:0) X n Z + a X n Y + a n +1 Y n +1 (cid:1) α = 0 Y ( n +1) α +1 + X (cid:0) X n Z + a X n Y + a n +1 Y n +1 (cid:1) α = 0 where n ≥ , α ≥ , a, a ∈ Q , and a n +1 ∈ ( Q ) ∗ . Then integral points on V are potentially dense. roof. We first prove part (i). Using the notation of Theorem 31, let τ = τ −→ a s ◦ · · · ◦ τ −→ a when s ≥ τ = id when s = 0. Let Λ be the pencil generatedby F ( τ ( x, y, z )) and F ( τ ( x, y, z )), where F i are chosen to be one of the threepossibilities stated in Theorem 31 (ii). By construction, the base points of Λare on D . Suppose at first that s ≥
1, i.e. some De Jonqui´eres transformationis necessary. Since the x -coordinate of τ ( x, y, z ) is a pure power of x with powerat least 2 and since α > α >
1, straightforward consideration of cases givesthat the Campana weight of ( D, Λ) is at least 1 + (1 − ) + (1 − ) >
2, unless( s, α , α , n, m ) = (1 , , n + 1 , n,
1) and D is defined by (11). So Theorem 23shows Zariski-non-density of integral points except for this case. In this specialcase, the Campana weight is exactly 2. Let τ = ( x , xz + a xy + a y , xy ) and let D be defined by ( x n z + P n +1 j =1 b j x n +1 − j y j ) + xy n +1 , where a , b n +1 ∈ ( Q ) ∗ .Then D is defined by x n +1 y + n +1 X j =1 b j x n +1 − j ) ( xz + a xy + a y ) j + x ( xz + a xy + a y ) n +1 = 0 . By replacing z + a y by z , the equation simplifies to x n +1 y + n +1 X j =1 b j x n +1 − j ) ( xz + a y ) j + x ( xz + a y ) n +1 , yielding the first equation in (i).We are now left with the case s = 0. For (10) and (12), when the power of x in F is at least 2, as α ≥
3, the Campana weight is greater than 2 and we mayapply Theorem 23. For (11), when α = 2, the gcd condition forces the powerof x in F to be odd, so the minimum power in F cannot be 2. Therefore, inthe case (11), the Campana weight is greater than 2 unless, again, the power of x in F is exactly equal to 1. Thus, the cases where the exponent of x in F isequal to 1 yield the other three possibilities in (i). Note that in each of thesecases the Campana weight of ( D, Λ) is less than 2.We now prove (ii). For the first case, we can use Lemma 25 with the linedefined by z = by for some b = − a . For the last two cases, we can use Lemma25 with the line defined by z = − a y . Remark 36.
Note that we need to use the Campana weight, rather than thegcd weight, to eliminate many cases in the proof of Theorem 35 (i). Since themember of Λ corresponding to F in (10)–(12) has at least two components, thegcd multiplicity of this member may be significantly smaller than the Campanamultiplicity, resulting in many cases where the the Campana weight is greaterthan two, but the gcd weight is not. In contrast to the previous case, we prove an unconditional result for rationalunicuspidal curves. 23 heorem 37.
Suppose that D is a rational curve defined over Q having exactlyone cusp and no other singularities and that V = P \ D satisfies ¯ κ ( V ) = 1 . Ifintegral points on V are potentially dense then D is projectively equivalent to (( X Z + Y ) Y + aX ) − ( X Z + Y ) X = 0 for some a ∈ Q ∗ .Proof. We use Theorem 32. We begin with the first case of that theorem. Let D be defined by f s − y + s X i =2 a i f s − i x ( n +1) i − n ! µ A − f µ G ! /x n = 0 , where f = x n z + y n +1 , µ A = n + 1, µ G = ( n + 1)( s −
1) + 1, and n, s ≥ (cid:0) f s − y + P si =2 a i f s − i x ( n +1) i − n (cid:1) µ A and f µ G .Since the divisor D occurs in the same member of Λ as x n = 0, this membercontributes a gcd multiplicity of n . So the gcd weight of ( D, Λ) is at least(1 − µ A ) + (1 − µ G ) + (1 − n ). When n ≥
3, we have µ A ≥ µ G ≥ D, Λ) is at least + + >
2. In addition, if n = 2and s ≥
3, then µ A = 3 and µ G ≥
7, so the gcd weight of ( D, Λ) is at least + + = >
2. Therefore, the only situation where ( D, Λ) possibly hasgcd weight ≤ n = s = 2. In this situation, the divisor is projectivelyequivalent to (cid:0) ( x z + y ) y + ax (cid:1) − ( x z + y ) x = 0for some a ∈ ( Q ) ∗ . Since every member other than the two multiple fibers andthe reducible fiber containing D is irreducible and reduced [27, Theorem 1], thegcd-weight (as well as the Campana weight) of this ( D, Λ) is + + = < D be defined by(( g n y + x n +1 ) µ A − ( g n z + 2 x n yg n + x n +1 ) µ G ) /g n = 0 , where g = xz − y , µ A = 4 n + 1 ≥
9, and µ G = 2 n + 1 ≥
5. Let Λ be thepencil generated by ( g n y + x n +1 ) µ A and ( g n z + 2 x n yg n + x n +1 ) µ G . Sincethe divisor D occurs in the same member of Λ as g n = 0, this member has gcdmultiplicity n . Therefore, the gcd weight of ( D, Λ) is at least + + >
2. Inthe third case, µ A = 4 n + 1 ≥ µ G = 2((4 n + 1) s − n ) ≥ n + 1) ≥ s ≥
1. By the same argument (with the natural choice of Λ), the gcd weightof ( D, Λ) is at least + + >
2. Applying Theorem 23 (b) finishes theproof. ≤ with a uniquesingularity Under certain conditions on the singularities, we prove potential density of in-tegral points on the complements of singular quartic and quintic curves.24 heorem 38.
Suppose that D is a quartic plane curve defined over Q witha triple point, i.e. a singularity of multiplicity three. Then integral points on P \ D are potentially dense.Proof. Let Λ be the pencil of lines in P passing through the singular point of D . Then for a general member C ∈ Λ, C \ D ∼ = G m over ¯ Q . By [16, Th. 3.5], D has four bitangent lines, whose eight points of contact lie on a conic. Thentaking C to be one of the bitangents and applying Lemma 25 to Λ and C , weconclude that integral points on P \ D are potentially dense. Theorem 39.
Suppose that D is a quintic rational plane curve defined over Q with a unique singular point, which has multiplicity with points above it inthe normalization. Then integral points on P \ D are potentially dense.Proof. We may assume that the singular point is [0 : 0 : 1]. By the genus–multiplicity formula (cf. [17, Example V.3.9.2]), the singularity of D is resolvedafter one blowup. Since the preimage of the singular point in the normalizationconsists of 3 points, the defining equation of D is F ( X, Y, Z ) = F ( X, Y ) + L ( X, Y ) L ( X, Y ) L ( X, Y ) Z, where F is a homogeneous polynomial of degree 5 in two variables and L , L , L are distinct linear polynomials. Without loss of generality, we may assumethat L = X and L = Y . Then for a suitable choice of a and b in Q , F ( X, Y ) − ( aX + bY ) is divisible by XY . We let G be the cubic polyno-mial G ( X, Y, Z ) = F ( X, Y, Z ) − ( aX + bY ) XY = F ( X, Y ) − ( aX + bY ) XY + L ( X, Y ) Z. The curve C defined by G = 0 has a cusp, and D ∩ C = { aX + bY = 0 } ∩ C ,so it follows that C \ D is isomorphic to G m . We also note that a general linethrough [0 : 0 : 1] meets D in two points, and so we conclude from Lemma 25that integral points on P \ D are potentially dense. Under the abc conjecture, we partially classify integral points on the comple-ment of a line and a plane curve when the complement has logarithmic Kodairadimension one.
Theorem 40.
Suppose that D is a union of a line and an irreducible curvedefined over Q and let V = P \ D . Assume the abc conjecture. If integralpoints on V are potentially dense and ¯ κ ( P \ D ) = 1 then V ∼ = P \ D ′ , where D ′ is defined by one of the following two equations:(i) Z ( Z b ( l +1) + X ( X l Y + p ( X, Z ) Z l +1 − deg p ) b ) = 0 , where b > , l > , deg p < l , and p (0 , = 0 . ii) Z ( a ( X, Z ) Y + a ( X, Z ) Z a − deg a ) = 0 , where a and a do nothave a common factor, deg a < deg a , and a has at least two distinctfactors.Proof. Let L be the line defined by Z = 0 and identify A = P \ L . If D is a sumof L and a curve C and φ : A → A is an automorphism, then integral pointson P \ D = A \ C will be transformed via φ to integral points on A \ φ ( C ∩ A ).Thus, we may reduce to analyzing the four cases classified by Aoki in Theorem34. Note that since Z = 0 is a part of the divisor, any multiplicity of Z isignored in computing the gcd/Campana weights.Suppose first that f ( x, y ) = x a y b + 1, where gcd( a, b ) = 1 and a > b >
1. Letting Λ be the pencil defined by X a Y b and Z a + b , the Campana weightof ( D, Λ) is at least (cid:16) − a,b ) (cid:17) + 1 + 1 >
2. So according to Theorem 23(assuming the abc conjecture), integral points are always Zariski-non-dense.Suppose now that f ( x, y ) = x a ( x l y + p ( x )) b + 1, where a > b > l > a, b ) = 1, deg p ( x ) < l and p (0) = 0. Letting Λ be the pencil generated by Z a + b ( l +1) and the homogenization of x a ( x l y + p ( x )) b , the Campana weight of( D, Λ) is at least (cid:16) − a,b ) (cid:17) + 1 + 1. This is greater than 2 if and only if a >
1. Then by Theorem 23, integral points are Zariski-non-dense unless a = 1.The case a = 1 yields the first possibility in the theorem.The second possibility in the theorem is the third case in Theorem 34, towhich we have nothing to add.Finally, suppose that f ( x, y ) = x a − y b with a > b >
1, and gcd( a, b ) = 1.Without loss of generality, let us assume that a < b , and let Λ be the pencilgenerated by X a Z b − a and Y b . Then the gcd/Campana weight of ( D, Λ) is atleast (cid:18) − a (cid:19) + (cid:18) − b (cid:19) + 1 ≥
12 + 23 + 1 > , since gcd( a, b ) = 1 and a, b >
1. Then by Theorem 23, integral points areZariski-non-dense (without assuming the abc conjecture).
Remark 41.
Here we list the cases of ¯ κ ( P \ D ) = 1 for which we have beenunable to classify integral points (even under the abc -conjecture). As in Remark30, a part of the problem is the lack of a classification theory in certain cases.As we will see in Lemma 43, integral points on P \ D are never Zariski-dense if D has at least four components. Thus, the unresolved cases due to a lack of anappropriate classification are when D is irreducible of degree at least 6 havingexactly one singularity which is not a cusp, or when D is reducible and eitherthe number of components is exactly 3 or none of the components is a line.For the rest of the unresolved cases, we have been unable to construct aZariski-dense set of integral points (or prove non-density) even though at leastsome classification result is known: 26i) the bicuspidal case which is projectively equivalent to one of X n +1 Y + n +1 X j =1 b j X n +1 − j ) ( XZ + cY ) j + X ( XZ + cY ) n +1 = 0 ,XY ( n +1) α − + X n Z + n +1 X j =1 a j X n +1 − j Y j α = 0 ,Y ( n +1) α +1 + X X n Z + n +1 X j =1 a j X n +1 − j Y j α = 0 , where n ≥ α ≥ a j , b j ∈ Q , at least one of a , . . . , a n is nonzero, and a n +1 .b n +1 , c ∈ ( Q ) ∗ .(ii) the unicuspidal case which is projectively equivalent to(( X Z + Y ) Y + aX ) − ( X Z + Y ) X = 0for some a ∈ Q ∗ .(iii) a rational plane curve of degree 5 with a unique singularity which hasmultiplicity e and e points above it in the normalization, e = 3 , Z ( Z b ( l +1) + X ( X l Y + p ( X, Z ) Z l +1 − deg p ) b ) = 0 ,Z ( a ( X, Z ) Y + a ( X, Z ) Z a − deg a ) = 0 , where b > l >
0, deg p < l , and p (0 , = 0 for the first type, and where a and a do not have a common factor, deg a < deg a , and a has atleast two distinct factors for the second type.The examples in Section 7 will show that there are some special examples of thecases listed above for which we can still conclude potential density of integralpoints. To prove Theorem 9, we will actually prove the following refinement, whichdescribes in more detail the situation when integral points in an orbit are Zariski-dense in P : 27 heorem 42. Let φ : P → P be a morphism of degree d > defined over anumber field k . Let D be a nontrivial effective divisor on P defined over k . Let S be a finite set of places of k containing the archimedean places. Let r = max { of distinct irreducible components of Supp( φ n ) ∗ D over ¯ k | n ∈ N } . (i) Suppose that there exists P ∈ P ( k ) such that O φ ( P ) contains a Zariskidense set of ( D, S ) -integral points on P . Then r ≤ .(ii) In the situation of (i), fix n such that Supp( φ n ) ∗ D has r irreducible com-ponents over ¯ k , and let C be an irreducible component of Supp( φ n ) ∗ D over ¯ k . Then for i ∈ N , C i = φ − i ( C ) is a geometrically irreducible curve,and assuming Lang–Vojta’s conjecture (Conjecture 1) for P , one of thefollowing two conditions is satisfied:(a) C is a line and φ − m ( C ) = C (as a set) for some positive integer m .(b) each C i is a rational curve, and there exist a positive integer i anda sequence of points P i , P i +1 , P i +2 , . . . such that for all i ≥ i , deg C i +1 = d deg C i , P i is a singular point of C i , and φ − ( P i ) = { P i +1 } (as a set). Further, the set { P i : i ≥ i } is a finite set. We note that as an immediate corollary, if r as above is at least 4, a setof ( D, S )-integral points in any orbit is Zariski-non-dense in P . If case (a) of(ii) holds for every irreducible component of Supp( φ n ) ∗ D , then after replacing k by a finite extension, D consists of r distinct lines L , . . . , L r , 1 ≤ r ≤
3, over k , and there is an integer m such that φ − m ( L i ) = L i for i = 1 , . . . , r . In fact,since ∪ ri =1 ∪ mj =1 φ − j ( L i ) is a completely invariant set, it is clear that we can take m ∈ { , , } by Lemma 46 below. If r = 3, then the lines L , L , L must be ingeneral position (otherwise, one can project from P \ D to P \ { three points } and apply Siegel’s theorem to deduce that integral points on P \ D are notZariski-dense). Then after possibly replacing k by a finite extension again andup to an automorphism of P over k , φ and D have one of the following forms:(1) D is the line z = 0 and φ = [ F ( X, Y, Z ) : G ( X, Y, Z ) : Z d ]for some homogeneous polynomials F ( X, Y, Z ) , G ( X, Y, Z ) ∈ k [ X, Y, Z ] ofdegree d .(2) D is defined by yz = 0 and φ = [ F ( X, Y, Z ) : Y d : Z d ]for some homogeneous polynomial F ( X, Y, Z ) ∈ k [ X, Y, Z ] of degree d .(3) D is defined by xyz = 0 and φ = [ X d : Y d : Z d ] .
28n case (3), each point in the orbit of [3 : 2 : 1] under φ is ( D, S )-integral for S = {∞ , , } , and it is easy to see from the 2-adic and the 3-adic valuations thatan algebraic curve can only contain finitely many points in this orbit. For cases(1) and (2), it follows from a recent work of Xie [35] that there exists an algebraicpoint whose orbit under φ is Zariski-dense (if there exists ℓ ≥ φ ℓ are multiplicativelyindependent, we can invoke [2, Corollary 2.7] instead). By enlarging k and S so that all coordinates of this point are S -units and that all the coefficients of F and G lie in O k,S , we see that each point in this orbit is ( D, S )-integral.Therefore, if case (b) of Theorem 42 never occurs, we have proved, under Lang–Vojta’s conjecture, a full generalization of Silverman’s theorem to P , completelycharacterizing endomorphisms of P and divisors D for which there exists anorbit of φ with a Zariski-dense set of ( D, S )-integral points.We also note that a general endomorphism does not have any totally ramifiedpoints. Even for an endomorphism with a totally ramified point, a generaldivisor D will not satisfy (a) or (b).We will first discuss several lemmata which will be used in the proof. Webegin by recalling a special case of a result of Vojta [30, Corollary 2.4.3]. Lemma 43.
Let S be a finite set of places of a number field k . Let D bean effective divisor on P n defined over k . If D has at least n + 2 irreduciblecomponents (over k ) then any set of ( D, S ) -integral points is not Zariski densein P n . More generally, using a reduction to unit equations, Vojta [30, Theorem 2.4.1]proved a result for an arbitrary nonsingular projective variety X , replacing thequantity n + 2 in the lemma by dim X + ρ + r + 1, where ρ is the Picard numberof X and r is the rank of Pic ( X )( k ). Using results on integral points on closedsubvarieties of semiabelian varieties, this quantity was subsequently improvedby Vojta [31, Corollary 0.3] to dim X − h ( X, O X ) + ρ + 1.The next lemma is an immediate corollary of known results for the logarith-mic Kodaira dimension of the complement of an irreducible plane curve. Lemma 44.
Assume Lang–Vojta’s Conjecture for P . Let D = C be a geomet-rically irreducible curve in P defined over a number field k , and let S be a finiteset of places of k containing the archimedean places. If there exists a Zariskidense set of ( D, S ) -integral points, then either C is a nonsingular cubic curveor C is a rational curve with at most two singular points.Proof. By the Lang–Vojta conjecture, integral points are never Zariski-densewhen ¯ κ ( P \ C ) = 2. Therefore, the lemma follows from the result of Wak-abayashi (Theorem 29). Lemma 45.
Let φ be an endomorphism of a projective variety X , both definedover a number field k . Let D be a nontrivial effective divisor on X defined over k . Let S be a finite set of places of k containing the archimedean places. Supposethat P ∈ X ( k ) and O φ ( P ) contains a Zariski dense set of ( D, S ) -integral points n X . Then for any positive integer n , O φ ( P ) contains a Zariski dense set of (( φ n ) ∗ D, S ) -integral points on X .Proof. By functoriality of local and global height functions with respect to pull-backs by morphisms, we have X v ∈ M k \ S λ v (( φ n ) ∗ D, Q ) = h (( φ n ) ∗ D, Q ) − X v ∈ S λ v (( φ n ) ∗ D, Q ) + O (1)= h ( D, φ n ( Q )) − X v ∈ S λ v ( D, φ n ( Q )) + O (1)= X v ∈ M k \ S λ v ( D, φ n ( Q )) + O (1)for all Q ∈ X ( k ). Let R be a set of ( D, S )-integral points in O φ ( P ). Then bythe above calculation, the set R n = { Q ∈ O φ ( P ) | φ n ( Q ) ∈ R } is a set of (( φ n ) ∗ D, S )-integral points in O φ ( P ), and the result follows.The following is a theorem in holomorphic dynamics on CP : Lemma 46.
Let φ : P → P be a morphism (defined over C ). If C ⊂ P isa finite union of algebraic curves such that φ − ( C ) = C as sets, then C is aunion of three or fewer lines.Proof. Fornæss and Sibony [14, Proposition 4.2] show that such a curve musthave degree ≤
3, and then remove all non-linear possibilities, except for a conic.Cerveau and Lins Neto [9, Th´eor`em 2] remove the conic possibility.The final lemma we will use is a simple consequence of the chain rule:
Lemma 47.
Let φ : P → P be a morphism. If C ⊂ P is a curve and P is asingular point of C , then φ − ( C ) is singular at every point in φ − ( P ) .Proof of Theorem 42. We first note that the bound r ≤ r , it alsofollows immediately that each C i is geometrically irreducible. If deg C i +1 = d deg C i , then it must be that C i +1 is in the ramification locus of φ . If thishappens for infinitely many i , then since the ramification locus contains onlyfinitely many curves, we must have φ − m ( C j ) = C j for some j and some m (bytaking multiples, we may assume that m > j ). It follows that C = φ j ( C j ) ⊆ φ − ( m − j ) ( C j ), and since φ − ( m − j ) ( C j ) is irreducible, in fact we have equality.Therefore, φ − m ( C ) = C , and C is a line by Lemma 46. So we are in case (a).Hence we may now assume that deg C i +1 = d deg C i for all sufficiently large i .By Lemma 44 and Lemma 45, each such C i must be rational and have 1 or 2singular points. Choose i so that C i has the maximum number of singularpoints in the family of curves C i . Let P i be a singular point of C i . From the30efinition of C i and Lemma 47, φ − ( P i ) must consist of a single point P i +1 ,which is a singular point of C i +1 . Continuing inductively, we define points P i ∈ C i , i ≥ i , and we are in case (b). Finally, [3, Theorem 1] shows thatthere are at most 9 points in P over which the preimage set is a singleton. So { P i : i ≥ i } must be a finite set. Proof of Theorem 9.
If (a) of Theorem 42 occurs, then C is a completely invari-ant proper Zariski-closed subset of P , while if (b) occurs, then the finite set { P i , P i +1 , . . . } is a completely invariant proper Zariski-closed subset of P . Here we collect together a series of examples, some of which demonstrate ourtheorems, while others extend our results in certain special cases.We first discuss an example where ¯ κ ( P \ D ) = −∞ . Example 48.
Let D be Yoshihara’s quintic: the zero locus of F ( X, Y, Z ) =(
Y Z − X )( Y Z − X Z − XY ) + Y . Then ¯ κ ( P \ D ) = −∞ and, as in[23], the associated pencil Λ of Theorem 2 has two multiple fibers: 2 D and 5 E ,where E is the zero locus of G ( X, Y, Z ) =
Y Z − X . Then in the notation ofTheorem 2, D ′ = ( F G = 0), r = 2, and integral points on P \ D are potentiallydense. On the other hand, if C is another member of the pencil, say definedby F + G (which is irreducible over Q ), then the corresponding divisor D ′ isgiven by ( F G ( F + G ) = 0). So r = 3 and by Theorem 2, any set of integralpoints on P \ C is contained in an effectively computable (possibly reducible)curve. This last statement is presumably not obvious from a purely Diophantineanalysis approach, working na¨ıvely from equations involving F + G .Next, we provide an example of Theorem 23 (c). Example 49.
Let D be defined by Z ( Y Z − X ) = 0. As proved in [20],¯ κ ( P \ D ) = 1, and so we are in the situation of Theorem 40 (but we won’t needthe abc conjecture). The pencil Λ generated by Y Z and X has two base points,and for a general member C , C \ D is isomorphic to G m . Since Z = 0 is containedin D , the gcd/Campana weight of ( D, Λ) is at least 1 + + >
2. Therefore,Theorem 23 (c) shows unconditionally that the set of S -integral points on P \ D is Zariski-non-dense and effectively computable. In this particular case, we canalso prove this directly. An S -integral point in this case is asking for [ x : y : 1]with x, y ∈ O k,S such that y − x ∈ O ∗ k,S . Choosing representatives u , . . . , u ℓ of O ∗ k,S / ( O ∗ k,S ) , we see that such a point induces an integral point on one ofthe finitely many elliptic curves Y = X + u j . Therefore, any such pair ( x, y )lies in one of finitely many effectively-computable curves.An example that sits somewhere between Theorem 23 (a) and Theorem 23(b) is the following. 31 xample 50. Let D be the divisor defined by Y Z − X = 0. This is abicuspidal curve with ¯ κ ( P \ D ) = 1, so this also serves as an example of Theorem35 (i) (note that this is listed as (10) in Theorem 31 but it is not one of the curveslisted in Theorem 35 (i)). The Campana weight of the pencil generated by Y Z and X is (1 − , )+ +1 >
2, while the gcd-weight is (1 − , )+ +1 <
2. Therefore, upon assuming the abc conjecture, Theorem 23 (a) implies thearithmetically interesting statement that for any finite set of primes S in Z , theset { ( a, b ) ∈ Z : a a powerful number, b a fifth power, a − b not divisible by p / ∈ S } lies in a finite union of curves (in fact, in a finite union of lines passing throughthe origin). This example illustrates clearly the role of the abc conjecture inanalyzing integral points.We now give an example of Theorem 35 (ii). Example 51.
Let D be defined by F ( X, Y, Z ) = Y α +1 + X ( XZ + Y ) α = 0.Given a unit u ∈ O ∗ k,S and a natural number m , we have (cid:18) u − u αm , , u m − u − u αm (cid:19) ∈ O k,S , and F (cid:18) u − u αm , , u m − u − u αm (cid:19) = u. Assuming that O ∗ k,S is infinite, as u and m vary this yields a Zariski-dense setof S -integral points on the complement of D . Note that D belongs to the n = 1case of the third type listed in Theorem 35 (ii). When n ≥
2, the explicitdescription of integral points becomes more complicated, since our proof relieson Lemma 25, which in turn uses the results of [1].We now discuss an example of a bicuspidal curve for which potential densityof integral points can be shown unconditionally, but which is not covered inTheorem 35 (ii).
Example 52.
Let D be defined by the polynomial F ( X, Y, Z ) = Y b +1 + X ( X Z + aXY + Y ) b . This belongs to the fourth type in Theorem 35 (i).The pencil Λ generated by Y b +1 and X ( X Z + aXY + Y ) b induces a G m -fibration on P \ D and ( D, Λ) has gcd/Campana weight 1 + b b +1 + (1 − ) < a is a natural number. Then for each u ∈ O ∗ k,S , there exists t ∈ O ∗ k,S such that t b +1 − t b − a ( u − ≡ u − ) , (13)namely t = u a = (( u −
1) + 1) a . Therefore, all the coordinates of the point P u = (cid:18) u − t b , , t b ( t b +1 − t b − a ( u − u − (cid:19) are in O k,S and F ( P u ) = u . It follows that P u ∈ ( P \ D )( O k,S ), viewing P u in P ( k ), and similar to the proof of Lemma 25, Theorem 1.1 of [1] tells32s that for all but finitely many u , the member of the pencil containing P u contains infinitely many integral points with respect to D . Assuming that O ∗ k,S is infinite, by varying u , we conclude that ( P \ D )( O k,S ) is Zariski-dense.Similarly, a congruence relation can help us with some cases of the union ofa line and an irreducible curve (cf. Theorem 40). Example 53.
Let l = 2 and p ( X, Z ) = aX + Z with a ∈ N in the first equationof Theorem 40, so that the divisor D is defined by Z ( Z b + X ( X Y + ( aX + Z ) Z ) b ) = 0 . For each u ∈ O ∗ k,S , we choose t ∈ O ∗ k,S satisfying (13). Then as before, the point h u − t b , t b ( t b +1 − t b − a ( u − u − , i is an S -integral point with respect to D . Therefore,assuming that O ∗ k,S is infinite, the same argument as in Example 52 shows that( P \ D )( O k,S ) is Zariski-dense.Another example of a union of a line and an irreducible curve for whichpotential density of integral points can be shown unconditionally is the following. Example 54.
Let D be the union of a nodal cubic and a non-tangent line goingthrough the singular point, say D = ( Y ( Y Z − X − X Z ) = 0). After oneblowup, the boundary divisor becomes normal-crossings, and it is easy to seethat ¯ κ ( P \ D ) = 1. Interchanging the roles of Y and Z , the affine equationon A is (1 − x )( y + x ) − x = 0, so this is case (iii) of Theorem 34 and thesecond possibility in Theorem 40. Going back to the original notation, any linethrough the singular point [0 : 0 : 1] other than Y = 0 and Y = ± X meets D atexactly one other point. So these lines form a pencil Λ such that for a generalmember C ∈ Λ, we have C \ D ∼ = G m over Q . We also have the line X = − Z which does not pass through the singular point, but meets D at only [ − C of Lemma 25.We end with an example for which we can unconditionally conclude Zariski-density of integral points, but which is not a part of Theorems 35–40. Example 55.
Let D be the sum of a conic and two non-tangent lines meeting ata common point P on the conic. For example, we can take the divisor D definedby ( X − Y )( Y Z − X ) = 0. By blowing up once at P , the boundary divisor D becomes a normal-crossings divisor, and it is easy to see that ¯ κ ( P \ D ) = 1.Any line through [0 : 0 : 1] meets D in just two points, yielding a pencil as inLemma 25. Moreover, the tangent line to the conic at [1 : 1 : 1] also meets D injust two points and this serves as the C of Lemma 25. For integral points on P \ D , the obvious question left to study is the remainderof the cases when ¯ κ ( P \ D ) = 1. For the cases listed in Remark 41, we have not33etermined whether integral points are potentially dense or not. We believe thatin these cases, the Campana weight of the pencil constructed from the structuretheory of affine surfaces will be less than or equal to 2. In some specific cases,such as the examples mentioned in Section 7, we can prove potential densityof integral points, but it seems one needs a more thorough classification theoryof affine surfaces and more involved arithmetic and Diophantine analyses toproceed further.One particular avenue of interest involves extending and generalizing thecongruence method of Examples 52 and 53. In these examples, when a is anatural number we were able to reduce the construction of a Zariski denseset of integral points to a congruence condition (13). For a general a ∈ Q ,it is possible that one may again be able to construct integral points using acongruence condition, rather than using a geometric method as in Lemma 25.For example, assume that k is a number field with a finite set of places S suchthat there are infinitely many u ∈ O ∗ k,S for which ( u − O k,S is a product ofprimes lying over distinct primes of Z S that split completely in O k,S . Then foreach such u , there exists an N such that O k,S / ( u − O k,S ∼ = Z /N Z . Let a ∈ k .Enlarging S , we can assume that a ∈ O k,S . Then there exist a natural number m and α ∈ O k,S such that a = m + α ( u − t b +1 − t b − a ( u − ≡ t b +1 − t b − m ( u − ≡ u − ) , and we can argue as in Example 52 to construct a Zariski-dense set of integralpoints. As a case in point, if p = 2 n − n ≥
3, is a (rational) prime, then p splits completely in O Q ( √ and ( √ n − O Q ( √ lying over p . So the assumption made above is satisfied for k = Q ( √
2) if thereare infinitely many Mersenne primes (taking S to consist of the archimedeanplaces and the place above the 2-adic place).As for integral points in orbits, we would like to further refine Question8. From [36, Proposition 15], one can have an endomorphism of P with acompletely invariant singleton set and yet integral points in every orbit areZariski non-dense in P . Therefore, it is natural to ask: for endomorphisms on P , can one conclude that there must be a one-dimensional completely invariantZariski-closed subset instead of just a nonempty completely invariant properZariski-closed subset in Question 8? This is related to determining whether (b)of Theorem 42 (ii) can actually occur. Acknowledgments
The authors would like to thank the Centre International de Rencontres Math´ematiques(CIRM) in Luminy and the organizers of the conference “Autour des conjecturesde Lang et Vojta”, which took place there, for providing the opportunity forstimulating discussions that formed the foundation of this work.34 eferences [1] Paraskevas Alvanos, Yuri Bilu, and Dimitrios Poulakis,
Characterizing al-gebraic curves with infinitely many integral points , Int. J. Number Theory (2009), no. 4, 585–590. MR 2532274 (2010f:11103)[2] Ekaterina Amerik, Fedor Bogomolov, and Marat Rovinsky, Remarks onendomorphisms and rational points , Compos. Math. (2011), no. 6,1819–1842. MR 2862064[3] Ekaterina Amerik and Fr´ed´eric Campana,
Exceptional points of an endo-morphism of the projective plane , Math. Z. (2005), no. 4, 741–754. MR2126212 (2005k:14021)[4] Hisayo Aoki, ´Etale endomorphisms of smooth affine surfaces , J. Algebra (2000), no. 1, 15–52. MR 1749875 (2001g:14092)[5] Frits Beukers,
Ternary form equations , J. Number Theory (1995), no. 1,113–133. MR 1352640 (96i:11028)[6] Yuri F. Bilu, Quantitative Siegel’s theorem for Galois coverings , CompositioMath. (1997), no. 2, 125–158. MR 1457336 (98d:11071)[7] Enrico Bombieri and Walter Gubler,
Heights in Diophantine geometry ,New Mathematical Monographs, vol. 4, Cambridge University Press, Cam-bridge, 2006. MR 2216774 (2007a:11092)[8] Fr´ed´eric Campana,
Orbifolds, special varieties and classification theory ,Ann. Inst. Fourier (Grenoble) (2004), no. 3, 499–630. MR 2097416(2006c:14013)[9] Dominique Cerveau and Alcides Lins Neto, Hypersurfaces exceptionnellesdes endomorphismes de C P( n ), Bol. Soc. Brasil. Mat. (N.S.) (2000),no. 2, 155–161. MR 1785266 (2002h:32014)[10] Henri Darmon, Faltings plus epsilon, Wiles plus epsilon, and the generalizedFermat equation , C. R. Math. Rep. Acad. Sci. Canada (1997), no. 1,3–14. MR 1479291 (98h:11034a)[11] Henri Darmon and Andrew Granville, On the equations z m = F ( x, y ) and Ax p + By q = Cz r , Bull. London Math. Soc. (1995), no. 6, 513–543. MR1348707 (96e:11042)[12] Gerd Faltings, Diophantine approximation on abelian varieties , Ann. ofMath. (2) (1991), no. 3, 549–576. MR 1109353 (93d:11066)[13] ,
The general case of S. Lang’s conjecture , Barsotti Symposium inAlgebraic Geometry (Abano Terme, 1991), Perspect. Math., vol. 15, Aca-demic Press, San Diego, CA, 1994, pp. 175–182. MR 1307396 (95m:11061)3514] John Erik Fornæss and Nessim Sibony,
Complex dynamics in higher dimen-sion. I , Ast´erisque (1994), no. 222, 5, 201–231, Complex analytic methodsin dynamical systems (Rio de Janeiro, 1992). MR 1285389 (95i:32036)[15] Rajendra Vasant Gurjar and Masayoshi Miyanishi,
Affine lines on loga-rithmic Q -homology planes , Math. Ann. (1992), no. 3, 463–482. MR1188132 (94g:14018)[16] Joe Harris, Theta-characteristics on algebraic curves , Trans. Amer. Math.Soc. (1982), no. 2, 611–638. MR 654853 (83m:14022)[17] Robin Hartshorne,
Algebraic geometry , Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics, No. 52.[18] Marc Hindry and Joseph H. Silverman,
Diophantine geometry , GraduateTexts in Mathematics, vol. 201, Springer-Verlag, New York, 2000, An in-troduction. MR 1745599 (2001e:11058)[19] Shigeru Iitaka,
Algebraic geometry , Graduate Texts in Mathematics, vol. 76,Springer-Verlag, New York-Berlin, 1982, An introduction to birational ge-ometry of algebraic varieties, North-Holland Mathematical Library, 24. MR637060 (84j:14001)[20] ,
Dais¯u kikagaku. I–III , second ed., Iwanami Shoten Kiso S¯ugaku[Iwanami Lectures on Fundamental Mathematics], vol. 23, Iwanami Shoten,Tokyo, 1984, Dais¯u [Algebra], vii. MR 845274 (87h:14001)[21] Yujiro Kawamata,
On the classification of noncomplete algebraic surfaces ,Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copen-hagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979,pp. 215–232. MR 555700 (81c:14021)[22] Hideo Kojima,
Structure of affine surfaces P − B with κ ≤
1, J. Algebra (2002), no. 1, 100–111. MR 1925009 (2003g:14082)[23] Masayoshi Miyanishi and Tohru Sugie,
On a projective plane curve whosecomplement has logarithmic Kodaira dimension −∞ , Osaka J. Math. (1981), no. 1, 1–11. MR 609973 (82k:14013)[24] Shinichi Mochizuki, Inter-universal Teichm¨uller theory IV: Log-volumecomputations and set-theoretic foundations , preprint available on his web-page, 2012.[25] Joseph H. Silverman,
Integer points, Diophantine approximation, and it-eration of rational maps , Duke Math. J. (1993), no. 3, 793–829. MR1240603 (95e:11070)[26] Keita Tono, Defining equations of certain rational cuspidal curves , 2000,Thesis (Ph.D.)–Saitama University.3627] ,
Rational unicuspidal plane curves with κ = 1,S¯urikaisekikenky¯usho K¯oky¯uroku (2001), no. 1233, 82–89, Newtonpolyhedra and singularities (Japanese) (Kyoto, 2001). MR 1905271[28] Shuichiro Tsunoda, The complements of projective plane curves ,S¯urikaisekikenky¯usho K¯oky¯uroku (1981), 48–55.[29] Machiel van Frankenhuijsen,
ABC implies the radicalized Vojta height in-equality for curves , J. Number Theory (2007), no. 2, 292–300. MR2362438 (2008j:11087)[30] Paul Vojta,
Diophantine approximations and value distribution theory , Lec-ture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR883451 (91k:11049)[31] ,
Integral points on subvarieties of semiabelian varieties. I , Invent.Math. (1996), no. 1, 133–181. MR 1408559 (98a:14034)[32] ,
A more general abc conjecture , Internat. Math. Res. Notices (1998),no. 21, 1103–1116. MR 1663215 (99k:11096)[33] ,
Integral points on subvarieties of semiabelian varieties. II , Amer.J. Math. (1999), no. 2, 283–313. MR 1680329 (2000d:11074)[34] Isao Wakabayashi,
On the logarithmic Kodaira dimension of the comple-ment of a curve in P , Proc. Japan Acad. Ser. A Math. Sci. (1978),no. 6, 157–162. MR 0498590 (58 The existence of Zariski dense orbits for polynomial endomor-phisms on the affine plane , preprint, arXiv:1510.07684, 2015.[36] Yu Yasufuku,
Deviation from S -integrality in orbits on P n , Bull. Inst. Math.Acad. Sin. (N.S.) (2014), no. 4, 603–631.[37] Hisao Yoshihara, On plane rational curves , Proc. Japan Acad. Ser. A Math.Sci. (1979), no. 4, 152–155. MR 533711 (80f:14016)Aaron LevinDepartment of MathematicsMichigan State University619 Red Cedar RoadEast Lansing, MI 48824USA [email protected] Yu YasufukuDepartment of MathematicsCollege of Science and Technology 37ihon University1-8-14 Kanda-Surugadai, Chiyoda-ku101-8308 TokyoJapan [email protected]@math.cst.nihon-u.ac.jp