An Evertse-Ferretti Nevanlinna constant and its consequences

In this paper, we introduce the notion of an Evertse-Ferretti Nevanlinna constant and compare it with the birational Nevanlinna constant introduced by the authors in a recent joint paper. We then use it to recover several previously known results. This includes a 1999 example of Faltings from his Baker's Garden article.


Introduction
Let X be a projective variety and let D be an effective Cartier divisor on X. Let L be a line sheaf on X with dim H 0 (X, L N ) > 1 for some N > 0. Other notations are as described in Section 3 and in [RV19].
In [RV19], the authors introduced the birational Nevanlinna constant Nev bir (L , D), which extends the notion of the Nevanlinna constant Nev(L , D) introduced by the first-named author [Ru17], and proved the following results.
Theorem A. Let k be a number field, and let S be a finite set of places of k containing all archimedean places. Let X be a projective variety over k, let D be an effective Cartier divisor on X, and let L be a line sheaf on X with dim H 0 (X, L N ) > 1 for some N > 0. Then, for every ǫ > 0, there is a proper Zariski-closed subset Z of X such that the inequality holds for all x ∈ X(k) outside of Z.
Theorem B. Let X be a complex projective variety, let D be an effective Cartier divisor, and let L be a line sheaf on X with dim H 0 (X, L N ) > 1 for some N > 0. Let f : C → X be a holomorphic mapping with Zariski-dense image. Then, for every ǫ > 0, (2) m f (r, D) ≤ exc (Nev bir (L , D) + ǫ) T f,L (r) .
Recall that Nev bir (L , D) and Nev(L , D) are defined as follows. (In [RV19] we defined them only for Q-Cartier divisors and required µ to be rational, but we are working over R in this paper. See Section 2 for more details on R-Cartier divisors.) Definition 1.1. Let X be a complete variety, let D be an effective R-Cartier divisor on X, and let L be a line sheaf on X. Then we define where the infimum passes over all triples (N, V, µ) such that N ∈ Z >0 , V is a linear subspace of H 0 (X, L N ) with dim V > 1, and µ ∈ R >0 , with the following property. There exist a variety Y and a proper birational morphism φ : Y → X such that the following condition holds. For all Q ∈ Y there is a basis B of V such that The Nevanlinna constant Nev(L , D) is defined similarly but without taking the birational model (i.e., it requires Y to be the normalization of X). Obviously, Nev bir (L , D) ≤ Nev(L , D).
The purpose of this note is to introduce the following variant of the Nevanlinna constant, based on a theorem of Evertse and Ferretti [EF08].
Definition 1.2. Let X be a complete variety, let D be an effective R-Cartier divisor on X, and let L be a line sheaf on X. Then we define where the infimum passes over all pairs (N, µ) such that N ∈ Z >0 and µ ∈ R >0 , with the following property. There exist a variety Y and a proper birational morphism φ : Y → X such that the following condition holds. For all Q ∈ Y there exist a base-point-free linear subspace V ⊆ H 0 (X, L N ) with dim V = dim X + 1 and a basis B of V such that If L is a Cartier divisor or Cartier divisor class on X, then we define Nev EF (L, D) = Nev EF (O(L), D). We also define Nev EF (D) = Nev EF (D, D).
Our main result is as follows.
Main Theorem. Let X be a geometrically integral variety over a number field k, let D be an effective R-Cartier divisor on X, and let L be a line sheaf on X. Then Combining this with Theorem A and Lemma 3.9, we then have Theorem 1.3. Let k be a number field, and let S be a finite set of places of k containing all archimedean places. Let X be a projective variety over k, and let D be an effective R-Cartier divisor on X. Then, for every ǫ > 0, there is a proper Zariski-closed subset Z of X such that the inequality holds for all x ∈ X(k) outside of Z.
Corollary 1.4. Let X be a projective variety, and let D be an ample Cartier divisor on X. If Nev EF (D) < 1 then there is a proper Zariski-closed subset Z of X such that any set of D-integral points on X has only finitely many points outside of Z.
A similar theorem and corollary also hold in the case of holomorphic curves. Theorem 1.3 generalizes the original theorem of Evertse and Ferretti, and we show how to recover the latter theorem from Theorem 1.3 (see Theorem 4.1).
Some other applications are also given. This includes discussion of a 1999 class of examples obtained by Faltings [Fal02].
These examples consist of irreducible divisors D on P 2 for which P 2 \ D has only finitely many integral points over any number ring, and over any localization of such a ring away from finitely many places.
Faltings' paper is notable for two reasons. First, the divisor D is irreducible. Prior to the paper, the only divisors D on P 2 for which such statements were known were divisors with at least four irreducible components. The second reason is that (as noted at the very end of [Fal02]) the paper gives examples of varieties for which finiteness of integral points is known, yet which cannot be embedded into semiabelian varieties. Prior to the paper, the only varieties for which such finiteness statements were known, and which could not be embedded into semiabelian varieties, were moduli spaces of abelian varieties.
Faltings' construction was further explored by Zannier [Zan05]  The method of Faltings' paper is to apply the original "filtration method" of Faltings and Wüstholz [FW94], together with their probabilistic version of Schmidt's Subspace Theorem, to certain coherent ideal sheaves on a certaiń etale cover of P 2 \ D.
This use of ideal sheaves naturally suggested the use of "generalized Weil functions" of the second-named author [Voj96,§ 7], which, as noted in [RV19], are now more aptly called b-Weil functions. We first rephrased Faltings' proof in terms of b-Weil functions and replaced the work of Faltings-Wüstholz with the method of Evertse and Ferretti [EF08].
In an effort to express this modified proof using the Nevanlinna constant of the first-named author, we formulated the "birational Nevanlinna constant" Nev bir (L , D) (Definition 1.1). We found in [RV19] that this definition worked well for adapting Autissier's proof, but that it was not possible to apply it to Faltings' examples without involving the Evertse-Ferretti method. This was what led to the formulation of the "Evertse-Ferretti Nevanlinna constant" Nev EF (L , D) (Definition 1.2).
The outline of this paper is as follows. Section 2 introduces R-Cartier b-divisors on a complete variety X and develops some of their basic properties. Section 3 gives more information on the Evertse-Ferretti Nevanlinna constant, including some equivalent formulations of it, and proves the Main Theorem. Section 4 contains some brief applications of Theorem 1.3, including the earlier result of Evertse and Ferretti. Section 5 concludes this note by discussing a 1999 class of examples obtained by Faltings [Fal02].

R-Cartier b-divisors and their b-Weil functions
This section introduces some notation for divisors and b-divisors with coefficients in R. (Divisors with real coefficients were used in [RV19], but were not fully described there because they were added as a late change to the paper.) Let k be a field of characteristic zero. As in [RV19], a variety over k is an integral scheme, separated and of finite type over k. It is said to be complete if it is proper over k.
2.1. Q-Cartier and R-Cartier divisors. If X is a variety over k, then we let CDiv(X) denote the group of Cartier divisors on X. As in [Laz04,1.3.B], an R-Cartier divisor on X is an element of the group Thus, every R-Cartier divisor D on X occurs as a finite sum D = c i D i , with c i ∈ R and D i ∈ CDiv(X) for all i. An R-Cartier divisor D is effective if it can be written in the above form with c i ≥ 0 and D i effective for all i.
If X is normal, then CDiv(X) is canonically identified with a subgroup of the group Div(X) of Weil divisors on X, and by [Har77, II 6.3A] a Cartier divisor on X is effective if and only if it is effective as a Weil divisor. Defining CDiv Q (X), Div R (X), and Div Q (X) similarly to (7), we have that the map Div(X) → CDiv(X) induces natural maps Div Q (X) → CDiv Q (X) and Div R (X) → CDiv R (X). These too are injective (Q is a localization of Z, hence is flat over Z, and R is flat over the field Q, respectively). Moreover these maps again preserve the respective cones of effective divisors.
Since the maps Div(X) → Div Q (X) → Div R (X) are injective and respect the notion of effectivity, the maps CDiv(X) → CDiv Q (X) → CDiv R (X) have these same properties when X is normal.
If X is not normal, then an effective (integral) Cartier divisor on X remains effective in CDiv Q (X) and CDiv R (X), but the converse does not hold. For example, the Cartier divisor (y/x) on the cuspidal cubic curve y 2 = x 3 is not effective, but it is effective as an Q-Cartier or R-Cartier divisor.
2.2. B-divisors with coefficients in Z, Q, and R. B-divisors consider divisors not only on a given variety, but also on blowings-up of the variety. The prefix 'b' means birational.
Let X be a variety. Following [BdFF12], a model over X is a proper birational morphism π : X π → X, modulo isomorphism over X. This will be denoted X π . The Zariski-Riemann space of X is defined as and a Cartier b-divisor on a variety X is an element of the group CDiv(X) := lim − → π CDiv(X π ) .
Thus, each Cartier b-divisor on X comes from a Cartier divisor D on some model X π . A Cartier b-divisor is said to be effective if X π and D can be chosen such that D is effective.
An R-Cartier b-divisor on X is said to be effective if it comes from an effective R-Cartier divisor on some model X π . The group CDiv Q (X) of Q-Cartier b-divisors on X is defined similarly.
Assume briefly that the variety X is normal. By [RV19, Remark 2.3], a Cartier divisor on X is effective if and only if the corresponding Cartier b-divisor on X is effective. Therefore the map CDiv(X) → CDiv(X) (from Cartier divisors on X to Cartier b-divisors on X) is injective and preserves effectivity. Likewise, the maps CDiv(X) → CDiv Q (X) → CDiv R (X) are injective and preserve effectivity.
If X is not normal, then an effective Cartier divisor on X still remains effective as a Cartier b-divisor (or as a Q-Cartier or R-Cartier b-divisor), but not conversely. The latter is illustrated by the same example as before.
In contrast to (non-birational) Cartier divisors, though, a Cartier bdivisor on an arbitrary variety X is effective if and only if it is effective as a Q-Cartier or R-Cartier b-divisor. This is because we can replace X with its normalization without affecting any of these groups of b-divisors.
In [RV19,Prop. 4.10a], we showed that the group CDiv(X) is a latticeordered group; i.e., an ordered group in which every two elements have a least upper bound D 1 ∨ D 2 . This, in turn, relied on [RV19, Lemma 4.8], which stated that if D is a Cartier divisor on a variety Y , then there is a proper model π : Y ′ → Y such that π * D is a difference of effective Cartier divisors with disjoint supports. This lemma remains true for Q-Cartier divisors (by multiplying the divisor in question by a positive integer so as to clear denominators), but is no longer true for R-Cartier divisors. An example of the latter is the R-divisor αL 1 − βL 2 , where L 1 and L 2 are distinct lines on P 2 and α, β > 0 are real numbers such that α/β is irrational. Therefore CDiv(X) Q is a lattice-ordered group, but CDiv(X) R is not. As it turns out, though, this is not an issue here. In this paper, the only instances of taking the least upper bound or greatest lower bound of two b-divisors involve only integral Cartier b-divisors.
2.3. B-Weil functions for R-Cartier b-divisors. Now assume that k is either a number field or the field C, and that X is a complete variety over k.
Then a b-Weil function for a R-Cartier b-divisor on X can be defined by R-linearity, using the definition of b-Weil function for (integral) Cartier bdivisors, and these b-Weil functions can be used as usual to define proximity and counting functions. In addition, [RV19, Lemma 2.5] extends to R-Cartier divisors: Lemma 2.1. Let X be a complete variety, and let U 1 , . . . , U n be Zariskiopen subsets of X that cover X.
Proof. By taking a finite refinement of {U i }, we may assume that each U i is affine, and that Then λ D i is locally M k -bounded from below on U i (M k ) for all i. Indeed, by [Lan83, Ch. 10, Prop. 1.3], the function − j c j log f ij is locally M kbounded from below on U i (M k ) for all i; therefore so is λ D i since by definition of Weil function and Néron function the difference between the two functions is locally M k -bounded.
The proof then concludes as in [RV19] (using standard properties of M kbounded sets).
Proposition 2.2. Let X be a complete variety over a number field, let D be an effective R-Cartier divisor on X, let L be a line sheaf on X, let V be a linear subspace of H 0 (X, L ) with dim V > 1, and let µ ∈ R >0 . Consider the following conditions.
The proof is the same as in [RV19], except that Lemma 2.1 replaces [RV19, Lemma 2.5] and the step of multiplying by a positive integer n is omitted.
This section provides more basic properties of the Evertse-Ferretti Nevanlinna constant (Definition 1.2), and proves the Main Theorem.
3.1. Growth conditions. We first recall the definition of µ-b-growth in [RV19] which appeared in the definition of Nev bir (L, D). This can now be stated using R-Cartier divisors instead of Q-Cartier divisors.
. Let X be a complete variety, let D be an effective R-Cartier divisor on X, let L be a line sheaf on X, let V be a linear subspace of H 0 (X, L ) with dim V > 1, and let µ > 0 be a real number. We say that D has µ-b-growth with respect to V and L if there is a model φ : Zariski-open neighborhood U of Q, relative to the cone of effective Rdivisors on U . Also, we say that D has µ-b-growth with respect to V if it satisfies the above condition with L = O(D).
Basically, Nev bir (L , D), according to its definition, is the infimum of (dim V )/µ over all triples (N, V, µ) such that µ > 0 and N D has µ-b-growth with respect to V and L N .
Remark 3.2. Def. 4.12 of [RV19] requires X to be normal, but this is unnecessary since the definition is birational in nature.
Remark 3.3. If an effective R-Cartier divisor D has µ-b-growth with respect to V and L , then it also has µ ′ -b-growth with respect to V and L for all µ ′ < µ. Therefore, if D is Q-Cartier (or Cartier), then the value of Nev bir (L , D) according to Definition 1.1 coincides with the value according to [RV19,Def. 4.12], because one can require µ to be rational in Definition 1.1 without changing the value of Nev bir (L , D).
Here is an equivalent condition for µ-b-growth that will be needed for the proof of Proposition 3.12, which compares µ-b-growth with µ-EF-growth. This condition is a special case of [RV19,Prop. 4.13]. For a fuller list of equivalent conditions, see loc. cit.
Proposition 3.4. Let X be a complete variety, let D be an effective R-Cartier divisor on X, let L be a line sheaf on X, let V be a linear subspace of H 0 (X, L ) with dim V > 1, and let µ > 0 be a real number. Then D has µ-b-growth with respect to V and L if and only if the following condition is true.
We similarly rephrase the definition of the Evertse-Ferretti Nevanlinna constant Nev EF (L , D) in terms of a definition that corresponds to that of µ-b-growth, and is suitable for applying the work of Evertse and Ferretti [EF08].
Definition 3.5. Let X be a complete variety, let D be an effective R-Cartier divisor on X, let L be a line sheaf on X, and let µ > 0 be a real number. We say that D has µ-EF-growth with respect to L if there is a model φ : Y → X of X such that for all P ∈ Y there exist a base-point-free linear subspace V ⊆ H 0 (X, L ) with dim V = dim X + 1 and a basis B of V such that Zariski-open neighborhood U of P , relative to the cone of effective Rdivisors on U . Also, we say that D has µ-EF-growth if it satisfies the above Similarly to what was done with µ-b-growth, we have the following proposition.
Proposition 3.6. Let L and D be as in Definition 3.5. Then the Evertse-Ferretti Nevanlinna constant Nev EF (L , D) is the infimum of (dim X + 1)/µ over all pairs (N, µ) such that µ > 0 is real and N D has µ-b-growth with respect to L N .
Remark 3.7. Let L and D be as in Definition 3.5, and let µ and r be positive real numbers. Then rD has (µ/r)-EF-growth with respect to L if and only if D has rµ-EF-growth with respect to L , and Nev EF (L , rD) = r Nev EF (L , D) .
In regard to the definition of µ-EF-growth, we have the following equivalent statements (for notations, see [RV19]). These parallel the full list of [RV19,Prop. 4.13].
Proposition 3.8. Let X be a complete variety, let D be an effective R-Cartier divisor on X, let L be a line sheaf on X, and let µ > 0 be a real number. Then the following are equivalent.
The condition of (iv) holds for at least one place v.
Note that the join in (13) exists, because it involves only integral Cartier b-divisors. (Note that, in the latter lemma, one can allow µ to be real and D to be an R-Cartier divisor: when working with R-divisors in the proof, the step of multiplying by a positive integer n is unnecessary; also, the last few lines in the proof work equally well with R-Cartier divisors in place of Q-Cartier divisors.) To finish the proof, it will suffice to show that (i) implies (ii).
Assume that condition (i) is true. Let φ : Y → X be a model that satisfies the condition of Definition 3.5. By quasi-compactness of Y , we may assume that only finitely many triples (U, V, B) occur. Let (U 1 , V 1 , B 1 ), . . . , (U ℓ , V ℓ , B ℓ ) be those triples. We may assume that (B i ) is represented by a Cartier divisor E on Y . Then, for each i, relative to the cone of effective R-Cartier divisors on U i . This gives (13).

3.2.
Mumford's theory of degree of contact. The proof of the Main Theorem also relies on some results from Mumford's theory of degree of contact.
In this theory, we let k be a field of characteristic 0, let x 1 , . . . , x q be homogeneous coordinates on . , x q ]/I Y be the homogeneous coordinate ring of Y , and let c = (c 1 , . . . , c q ) ∈ R q ≥0 . Then (as is standard) for all m ≥ 0 we define where the maximum is taken over all sets of monomials x a 1 · · · x a H Y (m) whose residue classes modulo I Y form a basis for S(Y ) m = H 0 (Y, O(m)).
In Mumford's theory it is better to consider only varieties that are geometrically integral, because H 0 (Y, L ) is much harder to control when Y becomes reducible upon base change to a larger field. Fortunately, though, theorems about Zariski-dense sets of rational points are all vacuously true on (integral) varieties that are not geometrically integral, because of the following simple fact.
Lemma 3.9. Let Y be an (integral) variety over a field k of characteristic 0. If Y (k) is Zariski dense, then Y is geometrically integral.
Proof. Since Y reg is open and Zariski dense, it contains a rational point. The existence of such a point implies that Y is geometrically integral. Indeed, let P be such a point, and let k ′ be the algebraic closure of k in the function field K(Y ). Then O Y,P is regular, hence normal. Since k ′ is integral over k, O Y,P contains k ′ . Therefore its residue field contains k ′ , so k = k ′ is algebraically closed in K(Y ), and therefore Y is geometrically integral.
Lemma 3.10. Let k and q be as above, let Y ⊆ P q−1 k be a geometrically integral projective variety, and let E be an extension field of k. Let Y E = Y × k E, viewed as a projective variety in P q−1 E via the map obtained from Y ֒→ P q−1 k by base change. Then, for all c ∈ R q ≥0 and all m ≥ 0, and Proposition 3.11. Let k and q be as above, let Y ⊆ P q−1 k be a geometrically integral projective variety, and let n = dim Y . Assume that Y is not contained in any coordinate hyperplane of P q−1 . Then the inequality holds for all c ∈ R q ≥0 and all collections j 0 , . . . , j n of indices such that the linear system on Y generated by x j 0 , . . . , x jn is base point free. Here the implicit constant in O(1/m) depends only on Y .
Proof. By Lemma 3.10 we may assume that k is algebraically closed.
Let c and j 0 , . . . , j n be as in the statement of the theorem, and let m ∈ Z >0 . By the theory of degree of contact (see [ We claim that with the implicit constant depending only on Y . Pick j 0 , . . . , j n such that that c j 0 = max j c j and x j 1 /x j 0 , . . . , x jn /x j 0 form a transcendence base for K(Y ) over k. Then, for each m > 0, the monomials in x j 0 , . . . , x jn of total degree m are linearly independent in H 0 (Y, O(m)), so the set {x ℓ 0 j 0 · · · x ℓn jn : ℓ 0 + · · · + ℓ n = m} of such monomials can be extended to a basis of H 0 (Y, O(m)). From the definition of S Y (m, c), and the fact that H Y (m) = ∆ n! m n + O(m n−1 ), it then follows that This proves (20). Combining (19) and (20) then gives (18).

Proof of the Main Theorem.
We are now ready to prove the Main Theorem. This, in turn, reduces to the following Proposition, which compares the notions of µ-b-growth and µ-EF-growth.
Proposition 3.12. Let X be a geometrically integral variety over a number field k, let D be an effective R-Cartier divisor on X, let L be a line sheaf on X, and let µ > 0 be a real number. Assume that D has µ-EF-growth with respect to L . Then, for all ǫ > 0, there exist a positive integer m, a real number ν, and a linear subspace V ⊆ H 0 (X, L m ), such that mD has ν-b-growth with respect to V and L m , and such that Proof. Assume that D has µ-EF-growth with respect to L . Let V 1 , . . . , V ℓ ; B 1 , . . . , B ℓ ; λ B 1 , . . . , λ B ℓ ; and λ D be as in condition (iv) of Proposition 3.8.
Let {s 1 , . . . , s q } be the elements of i B i . These sections determine a morphism Φ : X → P q−1 . Let Y be the image. Since the s i are nonzero sections, Y is not contained in any coordinate hyperplane. Also note that Φ need not be a closed embedding; in fact, it may happen that dim Y < dim X. However, it is true that Y is geometrically integral. Let n = dim Y and ∆ = deg Y .

Fix a place v of k.
For each j = 1, . . . , q, choose a local Weil function λ (s j ),v for the divisor (s j ) at v. Since each (s j ) is effective, we may assume that each λ (s j ),v is nonnegative.
We will apply (18) with c = (λ (s 1 ),v (x), . . . , λ (sq),v (x)) for some x ∈ X(k v ). By (15), there is an index i such that where the implicit constant does not depend on x (or i). Write B i = {s j 0 , . . . , s jn }. Then where the implicit constant does not depend on x. Since there are only finitely many possible values for i, the constant may also be taken independent of i.

Thus (21) holds for sufficiently large m.
On the other hand, by the definition of S Y (m, c) and our choice of c, there are bases B 1 , . . . , B r of V and corresponding local Weil functions λ B 1 ,v , . . . , λ Br ,v such that for all x ∈ X(k v ). Thus, after pulling the bases back to V and the local Weil functions back to X, we see that S Y (m, c) equals the left-hand side of (11), and hence mD has ν-b-growth with respect to V and L m .
This proposition then leads quickly to the proof of the Main Theorem.
Proof of the Main Theorem. We may assume that Nev EF (L , D) < ∞ (otherwise there is nothing to prove).
Let ǫ > 0. By the definition of Nev EF , there is a pair (N, µ) with N ∈ Z >0 and µ ∈ Q >0 , such that N D has µ-EF-growth with respect to L N and such that dim X + 1 µ < Nev EF (L , D) + ǫ .
By Proposition 3.12, there exist a positive integer m, a rational number ν, and a linear subspace V ⊆ H 0 (X, L mN ) such that mN D has ν-b-growth with respect to V and L mN , and such that dim We then have and the proof concludes by letting ǫ go to zero.

Some applications
In this section, we recover some known results by computing Nev EF (L , D) and then applying Theorem 1.3.
Let X be a projective variety of dimension n over a number field k (the analytic case will be similar). Let D 1 , . . . , D q be effective Cartier divisors on X. In this section, such divisors will be said to be in general position if for all I ⊆ {1, . . . , q}, every irreducible component of i∈I Supp D i has codimension |I|.
Application 1. We assume that D 1 , . . . , D q are in general position on X. In addition, we assume that there exist an ample divisor A on X and positive integers d 1 , . . . , d q such that D i is linearly equivalent to d i A for all i = 1, . . . , q. By replacing D i with (d/d i )D i , where d = lcm{d 1 , . . . , d q }, we may assume that d i = 1 for all i. Let D = D 1 + · · · + D q . To compute Nev EF (A, D), we take N such that N A is very ample, so there is a morphism φ : X → P m and hyperplanes H 1 , . . . , H q in P m such that φ * H i = N D i for all i. Since D 1 , . . . , D q are in general position on X, for each point P ∈ X there are at most n = dim X divisors among {D 1 , . . . , D q } passing through P , so one may choose distinct i 0 , . . . , i n ∈ {1, . . . , q} such that D i 0 , . . . , D in includes all of the D j passing through P . Then φ * H j i , i = 0, . . . , n, (regarding H as a section H ∈ H 0 (P m , O P m (1))) forms a basis of a subspace V ⊂ H 0 (X, N A) with dim V = n + 1 (or, equivalently, the Qdivisors 1 N φ * H j i , i = 0, . . . , n, are actually integral divisors, and they define a basis of a subspace V ⊂ H 0 (X, A) with dim V = n + 1). By the condition on general position, this subspace is base point free. Furthermore, we have, for every irreducible component E of D with P ∈ E, Hence, from the definition, Nev EF (D) ≤ n + 1 .
Thus we recover the following important theorem of Evertse and Ferretti.
Theorem 4.1 (Evertse-Ferretti [EF08]). Let X be a projective variety over a number field k, and let D 1 , . . . , D q be effective Cartier divisors on X in general position. Let S ⊂ M k be a finite set of places. Assume that there exist an ample divisor A on X and positive integers d i such that D i is linearly equivalent to d i A for i = 1, . . . , q. Then, for every ǫ > 0, holds for all k-rational points outside a proper Zariski closed subset of X.
Application 2. We now only assume that D 1 , . . . , D q are in general position on X. Let A be an ample Cartier divisor on X. Denote by ǫ D j (A) the Seshadri constant of D j with respect to A, which is defined as Noticing that the statement in (24) involves ǫ, it suffices to prove the inequality q j=1 v∈S c j λ D j ,v (x) < (n + 1 + ǫ)h A (x) in place of (24), with rational c j close to ǫ D j (A) chosen such that A − c j D j is ample for all j. By passing to Q-Cartier divisors and replacing D j with c j D j we can further assume that c j = 1 for all j.
In this case, we compute Nev EF (A, D) for D = D 1 + · · · + D q . Take N such that qN A is very ample, N D j is integral for all j, and N (A − D j ) is base point free for all j. According to Heier and Levin (see [HL, §3]), there is a morphism φ : X → P m and hyperplanes H 1 , . . . , H q in P m such that φ * H j ≥ N D j for all j and φ * H 1 , . . . , φ * H q are in general position. Then, for every P ∈ X, one can find hyperplanes H j i , i = 0, . . . , n, with {i 0 , . . . , i n } ⊂ {1, . . . , q} such that the collection includes all φ * H j passing through P , φ * H j i ≥ N D j i for all i, and φ * H j i , i = 1, . . . , n, are in general position. Hence φ * H j i , i = 0, . . . , n, forms a basis of a base-point free subspace V ⊂ H 0 (X, N A) with dim V = n + 1. Furthermore, for every irreducible component E of D with P ∈ E, we have Thus Nev EF (A, D) ≤ n + 1.
Then, by Theorem 1.3, we recover the following theorem.
Theorem 4.2 (Heier-Levin [HL]). Let X be a projective variety of dimension n defined over a number field k. Let D 1 , . . . , D q be effective Cartier divisors on X in general position. Let S be a finite set of places of k. Let A be an ample Cartier divisor on X. Then, for ǫ > 0, there exists a proper Zariski-closed subset Z ⊂ X such that for all points x ∈ X(k) \ Z, where, for all j, ǫ D j (A) is the Seshadri constant of D j with respect to A.
This also works for subschemes in general position; see [HL].
Application 3. We only assume that D 1 , . . . , D q are in l-subgeneral position on X.
Recall the definition of l-subgeneral position: Let V be a projective variety and X ⊂ V be an irreducible subvariety of dimension n. Cartier divisors D 1 , . . . , D q on V are said to be in l-subgeneral position on X if for every choice J ⊂ {1, . . . , q} with #J ≤ l + 1, The important tool to deal with l-subgeneral position is the following result, which is originally due to Quang (see Lemma 3.1 in [Quang19]). Using this lemma, similar to the second case, we can prove that, under the assumptions that ǫ D j (A) = 1 for j = 1, . . . , q and that D 1 , . . . , D q are in l-subgeneral position on X, Nev EF (A, D) ≤ (l − n + 1)(n + 1).

Thus, we have
Theorem 4.4 (He-Ru [HR19]). Let k be a number field and let S be a finite set of places of k. Let X be a projective variety of dimension n over k. Let D 1 , . . . , D q be effective divisors on X in l-subgeneral position with l ≥ n.
Let A be an ample Cartier divisor on X. Then, for all ǫ > 0, there is a proper Zariski-closed subset Z ⊂ X such that the inequality holds for all points x ∈ X(k) \ Z, This also works for subschemes in l-subgeneral position, by blowing them up; see [HR19].

An Example of Faltings
In this section we use the notion Nev EF (D) to recover the proof of a class of examples of Faltings that appeared in his Baker's Garden article [Fal02]. Recall that these examples consist of irreducible divisors D on P 2 for which P 2 \ D has only finitely many integral points over O k,S , where O k,S is the localization of the ring of integers of a number field k away from a finite set S of places of k.
Note that none of the applications in Section 4 used the birational model in Definition 1.2. In this section, however, this model is essential.
The main result, Theorem 5.2, covers all of Faltings' examples, yet its proof follows rather directly from Corollary 1.4. Since the latter corollary relies on Schmidt's Subspace Theorem, it necessarily involves varieties that can be embedded into semiabelian varieties (actually G N m ). Thus, Faltings' examples can be viewed as examples where one adds components to the divisor D to obtain a divisor whose complement can be embedded into a semiabelian variety, in such a way that the resulting diophantine inequality is strong enough to give a useful inequality for the original divisor D.
In particular, the Shafarevich conjecture (on semistable abelian varieties over a given number field with good reduction outside of a given finite set of places, proved by Faltings in 1983) stands out as presently the only diophantine result with all the hallmarks of a result proved by Thue's method (ineffective, but with bounds on the number of counterexamples), but which has not been proved by Thue's method. This theorem amounts to showing finiteness of integral points on A g,n , which also has no embedding into a semiabelian variety. It would be interesting to know if some variant of the above approach could be used to derive finiteness of integral points on A g,n (and therefore the Shafarevich conjecture) by methods that ultimately rely on Thue's method.
In this section, we give a revised proof of Faltings' result. We will split Faltings' main result into a geometric part and an arithmetic part. The geometric part (Theorem 5.1) guarantees that examples with certain properties exist, while the arithmetic part (Theorem 5.2) says that in each such example P 2 \D has only finitely many integral points over O k,S . We prove only the arithmetic part here, since that is the part that involves the Evertse-Ferretti Nevanlinna constant.
The first (geometric) part of Faltings' result is stated as follows.
Theorem 5.1 (Faltings). Let k be a field of characteristic zero, and let X be a smooth geometrically irreducible algebraic surface over k. Then, for all sufficiently positive line sheaves L on X, there exists a morphism f : X → P 2 that satisfies the following conditions.
The ramification locus Z of f is smooth and irreducible, and the ramification index is 2. (iii) The restriction of f to Z is birational onto its image D ⊆ P 2 . (iv) D is nonsingular except for cusps and simple double points.
(v) Let Y → X → P 2 denote the Galois closure of X → P 2 (i.e., the normalization of X in the Galois closure of K(X) over K(P 2 )). Also let n = deg f . Then Y is smooth and its Galois group over P 2 is the full symmetric group S n . (vi) The ramification locus of Y over P 2 is the sum of distinct conjugate effective divisors Z ij , 1 ≤ i < j ≤ n. They have smooth supports, and are disjoint with the following two exceptions. Points of Y lying over double points of D are fixed points of a subgroup S 2 × S 2 of S n , and they lie on Z ij ∩ Z ℓm with distinct indices i, j, ℓ, m. Points of Y lying over cusps of D are fixed points of a subgroup S 3 of S n , and lie on Z ij ∩ Z iℓ ∩ Z jℓ .
For a proof of this theorem, and also an explicit description of the "sufficiently positive" condition on L , see Faltings' paper [Fal02].
For convenience, write Z ij = Z ji when i, j ∈ {1, . . . , n} and i > j. Let Let L be the divisor class of L on X, and let it also denote the pull-back of this divisor class to Y . In addition, let d = deg D. We then have Using the setup as above, we now use Evertse-Ferretti Nevanlinna constants together with Corollary 1.4 to prove the other part of Faltings' result.
Theorem 5.2. Let k be a number field and let S be a finite set of places of k. Let Y , n, {Z ij } i<j , {A i } i , and M be as in Theorem 5.1 and the discussion following it. Also let α be a rational number such that M − αA i is an ample Q-divisor for all i. Then: (a). if α > 6 then no set of O k,S -integral points on Y \ Z ij is Zariskidense, and (b). if α > 8 then every set of O k,S -integral points on Y \ Z ij is finite.
The first part of the proof of this theorem is the following proposition, which contains all of the geometry specific to the situation of Theorem 5.1.
Proposition 5.3. Let k be a number field, and let Y , n, {Z ij } i<j , {A i } i , M , and α be as in Theorem 5.2. Assume that n ≥ 4. Fix Weil functions λ ij for each Z ij . Let β be an integer such that βα ∈ Z and such that βM and all β(M − αA i ) are very ample. Fix an embedding Y ֒→ P N k associated to a complete linear system of βM , and regard Y as a subvariety of P N k via this embedding. Then holds for all v ∈ M k and for all but finitely many y ∈ C(k v ).
The proof of this proposition, in turn, relies mainly on two lemmas. These lemmas replace Faltings' computations of ideals associated to indices.
(a). there exist hyperplanes H 0 , H 1 , and H 2 in P N k , such that is an effective Cartier divisor on Y ; and (b). given any integral curve C ⊆ Y not contained in any of the Z ab , there are hyperplanes H 0 and H 1 in P N k , such that C ∩ H 0 ∩ H 1 = ∅ and is an effective Cartier divisor on C.
Proof. Let σ i and σ j be as in the preceding proof. Choose a section of the linear system σ βα i · Γ(Y, β(M − αA i )) + σ βα j · Γ(Y, β(M − αA j )) , and let H 0 be the associated hyperplane. Then H 0 Y − βαZ ij is effective. We may assume that the choice of H 0 is sufficiently generic so that H 0 does not contain any irreducible component of A ℓ .
Next let σ ℓ be the canonical section of O(A ℓ ), and let H 1 be the hyperplane associated to a section of σ βα ℓ · Γ(Y, β(M − αA ℓ )) . Then H 1 Y − βα(Z iℓ + Z jℓ ) is effective. We may assume that H 1 does not contain any irreducible component of Y ∩ H 0 .
Again, Y ∩ H 0 ∩ H 1 consists of finitely many points, and we choose H 2 to be any hyperplane not meeting any of these points. Part (a) then concludes as in the previous lemma.
Proof of Proposition 5.3. First consider part (a) of the proposition.
Fix a place v ∈ M k . Apply Lemmas 5.4a and 5.5a to all possible collections i, j, ℓ, m and i, j, ℓ of indices, respectively. This involves only finitely many applications, so only finitely many hyperplanes occur. Let H 1 , . . . , H q be those hyperplanes.
The conditions in Theorem 5.1 on the intersections of the divisors Z ij imply that there is a constant C v such that, for each y ∈ Y (k) not in Supp Z ij , one of the following conditions holds.
(i) λ ij (y) ≤ C v for all i and j; (ii) there are indices i and j such that λ ij (y) > C v but λ ab (y) ≤ C v in all other cases; (iii) there are distinct indices i, j, ℓ, m such that λ ij (y) > C v and λ ℓm (y) > C v but λ ab (y) ≤ C v in all other cases; or (iv) there are indices i, j, ℓ such that max{λ ij (y), λ iℓ (y), λ jℓ (y)} > C v , but λ ab (y) ≤ C v if {a, b} {i, j, ℓ}.
For case (iii), (25) follows from Lemma 5.4a, since one can take J corresponding to the hyperplanes occurring in the lemma, and the inequality will then follow from effectivity of (27). Case (ii) follows as a special case of this lemma, since n ≥ 4. Case (iv) follows from Lemma 5.5a, by a similar argument. Finally, in case (i) there is nothing to prove. This proves (a).
For part (b), let H 1 , . . . , H q be a finite collection of hyperplanes occurring in all possible applications of Lemmas 5.4b and 5.5b with the given curve C. We have cases (i)-(iv) as before. Cases (ii) and (iii) follow from Lemma 5.4b, where we may assume without loss of generality that λ ij (y) ≥ λ ℓm (y) to obtain (26) from effectivity of (28). Similarly, case (iv) follows from Lemma 5.5b after a suitable permutation of the indices, and case (i) is again trivial. In the proof, one would let φ : Y → X be a model for which the left-hand side of (30) pulls back to an ordinary divisor, and for each P ∈ X one would consider four cases: (i) P / ∈ Z ij for all i, j; (ii) P ∈ Z ij for exactly one pair i, j; (iii) there are distinct indices i, j, ℓ, m such that P ∈ Z ij and P ∈ Z ℓm , but P / ∈ Z ab for all other components; and (iv) there are indices i, j, ℓ such that P lies on at least two of Z ij , Z iℓ , and Z jℓ , but P / ∈ Z ab if {a, b} {i, j, ℓ}.
In each case let U be the complement of all Z ab that do not contain P . Then U is an open neighborhood of P in X, and there is a set J of indices such