A Schmidt–Nochka theorem for closed subschemes in subgeneral position

Gordon Heier; Aaron Levin

doi:10.1515/crelle-2024-0085

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A Schmidt–Nochka theorem for closed subschemes in subgeneral position

Gordon Heier and Aaron Levin

Published/Copyright: November 28, 2024

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 819

Abstract

In previous work, the authors established a generalized version of Schmidt’s subspace theorem for closed subschemes in general position in terms of Seshadri constants. We extend our theorem to weighted sums involving closed subschemes in subgeneral position, providing a joint generalization of Schmidt’s theorem with seminal inequalities of Nochka. A key aspect of the proof is the use of a lower bound for Seshadri constants of intersections from algebraic geometry, as well as a generalized Chebyshev inequality. As an application, we extend inequalities of Nochka and Ru–Wong from hyperplanes in 𝑚-subgeneral position to hypersurfaces in 𝑚-subgeneral position in projective space, proving a sharp result in dimensions 2 and 3, and coming within a factor of 3 / 2 of a sharp inequality in all dimensions. We state analogous results in Nevanlinna theory generalizing the second main theorem and Nochka’s theorem (Cartan’s conjecture).

1 Introduction

Schmidt’s subspace theorem occupies a central position in the theory of higher-dimensional Diophantine approximation (see [33, 34, 35, 36, 40] for its early forms). It has been generalized by many authors, including Evertse and Ferretti (see [10, 9, 11]), Corvaja and Zannier (see [7]), and Ru and Vojta [29]. In our recent paper [14], we proved the following version in terms of closed subschemes in general position and their Seshadri constants with respect to a given ample divisor 𝐴 (see also recent work of Ru and Wang [30, 31] and Vojta [43] involving beta constants in place of Seshadri constants; see [19] for a version involving Seshadri constants of points).

Theorem 1.1

Theorem 1.1 ([14, Theorem 1.3])

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘. Let 𝑆 be a finite set of places of 𝑘. For each v ∈ S , let Y 0 , v , … , Y n , v be closed subschemes of 𝑋, defined over 𝑘, and in general position. Let 𝐴 be an ample Cartier divisor on 𝑋, and ϵ > 0 . Then there exists a proper Zariski-closed subset Z ⊂ X such that, for all points P ∈ X ⁢ ( k ) ∖ Z ,

∑ v ∈ S ∑ i = 0 n ϵ Y i , v ⁢ ( A ) ⁢ λ Y i , v , v ⁢ ( P ) < ( n + 1 + ϵ ) ⁢ h A ⁢ ( P ) .

In the paper [13], we investigated the implications of this theorem for the degeneracy of integral points and entire curves in the complement of nef effective divisors, culminating in new (quasi-)hyperbolicity theorems under weak positivity assumptions based on seeking a certain lexicographical minimax. For earlier work on hyperbolicity, see [1, 2, 5, 6, 8, 17, 25, 41].

In the present paper, we extend Theorem 1.1 to the following Schmidt–Nochka-type theorem for weighted sums involving arbitrary closed subschemes (i.e., without any general position assumption).

Theorem 1.2

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘, and let 𝑆 be a finite set of places of 𝑘. For each v ∈ S , let Y 1 , v , … , Y q , v be closed subschemes of 𝑋, defined over 𝑘 (not necessarily in general position), and let c 1 , v , … , c q , v be nonnegative real numbers. For a closed subset W ⊂ X and v ∈ S , let

α v ⁢ ( W ) = ∑ i W ⊂ Supp ⁡ Y i , v c i , v .

Let 𝐴 be an ample Cartier divisor on 𝑋, and ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that, for all points P ∈ X ⁢ ( k ) ∖ Z ,

∑ v ∈ S ∑ i = 1 q c i , v ⁢ ϵ Y i , v ⁢ ( A ) ⁢ λ Y i , v , v ⁢ ( P ) < ( ( n + 1 ) ⁢ max v ∈ S ∅ ⊊ W ⊊ X ⁡ α v ⁢ ( W ) codim ⁡ W + ϵ ) ⁢ h A ⁢ ( P ) .

Our flexible use of weights in the above inequality appears to be completely novel and will be crucial in Section 6. Furthermore, Theorem 1.2 can be considered sharp by the following example.

Example 1.3

Fix a point P 0 ∈ P n ⁢ ( k ) and let r ≥ 1 be an integer. Consider hyperplanes H 1 , … , H r ⁢ n (not depending on v ∈ S ) which intersect at P 0 but otherwise intersect generally. Let 𝐻 be a hyperplane intersecting H 1 , … , H r ⁢ n generally and set H r ⁢ n + i = H , i = 1 , … , r , and c i , v = 1 for all 𝑖 and v ∈ S . Then it is easy to verify that the hyperplanes are in 𝑚-subgeneral position with m = r ⁢ n and

max v ∈ S ∅ ⊊ W ⊊ X ⁡ α v ⁢ ( W ) codim ⁡ W = m n = r ⁢ n n = r ,

where the maximum is attained at W = P 0 . Therefore, Theorem 1.2, with the ample divisor 𝐴 being any hyperplane and all Seshadri constants equal to 1, yields the trivial bound

∑ i = 1 r ⁢ ( n + 1 ) m H i , S ⁢ ( P ) < ( r ⁢ ( n + 1 ) + ϵ ) ⁢ h ⁢ ( P )

for all points P ∈ P n ⁢ ( k ) ∖ Z . Note that any line 𝐿 passing through P 0 and not contained in any of the hyperplanes H 1 , … , H r ⁢ n intersects H 1 , … , H r ⁢ ( n + 1 ) in at most two distinct points. Since such a line 𝐿 contains an infinite set of ( ∑ i = 1 r ⁢ ( n + 1 ) H i , S ) -integral points (and such lines are Zariski dense in P n ), we cannot replace r ⁢ ( n + 1 ) in the inequality above by anything smaller. This gives a family of examples where Theorem 1.2 is sharp.

We now explain the connections with the classical work of Schmidt and Nochka. To begin, Theorem 1.2 recovers a weighted closed subscheme version of many recent inequalities generalizing Schmidt’s subspace theorem, including work of Quang [26, 27, 28], Ji–Yan–Yu [15], and Shi [37]. In particular, if Y 1 , v , … , Y q , v are in general position and c i , v = 1 for all 𝑖 and 𝑣, we have by definition that codim ⁡ W ≥ α v ⁢ ( W ) for all v ∈ S and W ≠ ∅ , and we recover Theorem 1.1. We discuss in more detail the implications and relations with this previous work in Section 4.

On the other hand, to justify the reference to Nochka, we recall that, in 1982–1983, Nochka [21, 22, 23] proved a conjecture of Cartan on defects of holomorphic curves in P n relative to a possibly degenerate set of hyperplanes. Nochka’s work was further explained and simplified by Chen [4] and Ru and Wong [32], with the latter work proving versions of Nochka’s results in Diophantine approximation, including the following inequality for proximity functions associated to hyperplanes in 𝑚-subgeneral position.

Theorem 1.4

Theorem 1.4 (Ru–Wong [32])

Let H 1 , … , H q be hyperplanes of P n , defined over a number field 𝑘, and in 𝑚-subgeneral position. Let 𝑆 be a finite set of places of 𝑘 and let ϵ > 0 . Then there exists a finite union of hyperplanes Z ⊂ P n such that, for all points P ∈ P n ⁢ ( k ) ∖ Z ,

(1.1) ∑ i = 1 q m H i , S ⁢ ( P ) < ( 2 ⁢ m − n + 1 + ϵ ) ⁢ h ⁢ ( P ) .

It is known that the quantity 2 ⁢ m − n + 1 in the inequality is sharp for all values of m ≥ n (for an appropriate configuration of hyperplanes). A key inequality in Ru and Wong’s work is the following inequality involving a sum of local heights weighted by Nochka weights (the analogous inequality in Nevanlinna theory plays a similar key role in Nochka’s proof of Cartan’s conjecture).

Theorem 1.5

Theorem 1.5 (Ru–Wong [32])

Let 𝑆 be a finite set of places of a number field 𝑘, and for each v ∈ S , let H 1 , v , … , H q , v be hyperplanes of P n , defined over 𝑘 and in 𝑚-subgeneral position, with associated Nochka weights ω 1 , v , … , ω q , v . Let ϵ > 0 . Then there exists a finite union of hyperplanes 𝑍 of P n such that, for all points P ∈ P n ⁢ ( k ) ∖ Z ,

∑ v ∈ S ∑ i = 1 q ω i , v ⁢ λ H i , v , v ⁢ ( P ) < ( n + 1 + ϵ ) ⁢ h ⁢ ( P ) .

A fundamental property of Nochka weights is that, for a nonempty closed subset W ⊂ X and v ∈ S ,

∑ i W ⊂ H i , v ω i , v ≤ codim ⁡ W .

Thus, we see that Theorem 1.5 is a consequence of Theorem 1.2 (with c i , v = ω i , v , Y i , v = H i , v ), excluding the linearity of the exceptional set 𝑍. Then we may view Theorem 1.2 as jointly generalizing Schmidt’s subspace theorem as well as inequalities arising out of Nochka’s work and its extensions.

Just as Theorem 1.5 is a key ingredient in proving Ru–Wong’s inequality (1.1), we derive from Theorem 1.2 a version of inequality (1.1) for arbitrary effective divisors, under a Bezout-type intersection assumption which is, in particular, valid for projective space.

Theorem 1.6

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘 and let 𝑆 be a finite set of places of 𝑘. Let D 1 , … , D q be effective Cartier divisors on 𝑋, defined over 𝑘, in 𝑚-subgeneral position. Let D I = ⋂ i ∈ I D i . We assume the following Bezout property holds for intersections among the divisors: if I , J ⊂ { 1 , … , q } , then

codim ⁡ D I ∪ J = codim ⁡ ( D I ∩ D J ) ≤ codim ⁡ D I + codim ⁡ D J .

Let 𝐴 be an ample divisor on 𝑋 and let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that, for all P ∈ X ⁢ ( k ) ∖ Z ,

(1.2) ∑ i = 1 q ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) < ( 3 2 ⁢ ( 2 ⁢ m − n + 1 ) + ϵ ) ⁢ h A ⁢ ( P ) .

As is well known, the Bezout property holds when X = P n , and if D 1 , … , D q are hypersurfaces in P n of degrees d 1 , … , d q ≥ 1 , and h A = h is the standard height (associated to 𝐴 a hyperplane), we have ϵ D i ⁢ ( A ) = 1 d i and inequality (1.2) becomes (outside some proper closed subset of P n )

∑ i = 1 q 1 d i ⁢ m D i , S ⁢ ( P ) < ( 3 2 ⁢ ( 2 ⁢ m − n + 1 ) + ϵ ) ⁢ h ⁢ ( P ) .

Thus, we obtain a version of (Nochka–)Ru–Wong’s Theorem 1.4 with an extra factor of 3 2 , but valid for arbitrary hypersurfaces in projective space. This inequality does not seem to have been previously known (for all 𝑚 and 𝑛) even with 3 2 replaced by an arbitrarily large constant; an inequality with a factor of ( m − n + 1 ) ⁢ ( n + 1 ) + ϵ on the right-hand side follows from work of Quang [26].

In low dimensions ( n ≤ 3 ), we are able to remove the factor 3 2 and prove a full generalization of Ru–Wong’s theorem to hypersurfaces in projective space.

Theorem 1.7

Let n ≤ 3 be a positive integer. Let D 1 , … , D q be effective divisors on P n , defined over 𝑘, in 𝑚-subgeneral position, of degrees d 1 , … , d q . Let ϵ > 0 and let 𝑆 be a finite set of places of 𝑘. Then there exists a proper Zariski-closed subset 𝑍 of P n such that, for all P ∈ P n ⁢ ( k ) ∖ Z ,

∑ i = 1 q 1 d i ⁢ m D i , S ⁢ ( P ) < ( 2 ⁢ m − n + 1 + ϵ ) ⁢ h ⁢ ( P ) .

In fact, we prove more general inequalities in dimension at most 3 under suitable geometric assumptions, and more generally for weighted sums of proximity functions (Theorems 6.3 and 6.4).

Using the well-known correspondence between statements in Diophantine approximation and Nevanlinna theory [39], the proof of Theorem 1.2 can be adapted to prove the following generalization of the second main theorem in Nevanlinna theory.

Theorem 1.8

Let 𝑋 be a complex projective variety of dimension 𝑛. Let Y 1 , … , Y q be closed subschemes of 𝑋 and let c 1 , … , c q be nonnegative real numbers. Let 𝐴 be an ample Cartier divisor on 𝑋 and let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that if f : C → X is a holomorphic map with f ⁢ ( C ) ⊄ Z , then

∫ 0 2 ⁢ π max J ⁢ ∑ j ∈ J c j ⁢ ϵ Y j ⁢ ( A ) ⁢ λ Y j ⁢ ( f ⁢ ( r ⁢ e i ⁢ θ ) ) ⁢ d ⁢ θ 2 ⁢ π ≤ exc ( Δ ⁢ ( n + 1 ) + ϵ ) ⁢ T f , A ⁢ ( r ) ,

where the maximum is taken over all subsets 𝐽 of { 1 , … , q } such that, for every nonempty closed subset W ⊂ X ,

∑ j ∈ J W ⊂ Supp ⁡ Y j c j ≤ Δ ⁢ codim ⁡ W .

The notation ≤ exc means that the inequality holds for all r ∈ ( 0 , ∞ ) outside of a set of finite Lebesgue measure. When c i = 1 for all 𝑖, this was proven (with a slightly different Δ) independently by Quang [28] (see the remarks on the analogous Diophantine result after Theorem 4.6).

Analogous to Theorems 1.6 and 1.7 and their proofs, we have the following consequences for holomorphic curves and proximity functions associated to divisors in 𝑚-subgeneral position. In particular, we extend Nochka’s theorem (Cartan’s conjecture) to hypersurfaces in projective space, however with an extra factor of 3 2 on the right-hand side of the inequality (and we eliminate this extra factor in dimensions at most 3).

Theorem 1.9

Let 𝑋 be a complex projective variety of dimension 𝑛. Let D 1 , … , D q be effective Cartier divisors on 𝑋 in 𝑚-subgeneral position. Let D I = ⋂ i ∈ I D i . Assume the following Bezout property holds for intersections among the divisors: if I , J ⊂ { 1 , … , q } , then

codim ⁡ D I ∪ J = codim ⁡ ( D I ∩ D J ) ≤ codim ⁡ D I + codim ⁡ D J .

Let 𝐴 be an ample Cartier divisor on 𝑋 and let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that if f : C → X is a holomorphic map with f ⁢ ( C ) ⊄ Z , then

∑ i = 1 q ϵ D i ⁢ ( A ) ⁢ m f , D i ⁢ ( r ) ≤ exc ( 3 2 ⁢ ( 2 ⁢ m − n + 1 ) + ϵ ) ⁢ T f , A ⁢ ( r ) ⁢ ( P ) .

In small dimensions, we have the better result.

Theorem 1.10

Let n ≤ 3 be a positive integer. Let D 1 , … , D q be effective divisors on P n in 𝑚-subgeneral position of degrees d 1 , … , d q . Let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of P n such that if f : C → P n is a holomorphic map with f ⁢ ( C ) ⊄ Z , then

∑ i = 1 q 1 d i ⁢ m f , D i ⁢ ( r ) ≤ exc ( 2 ⁢ m − n + 1 + ϵ ) ⁢ T f ⁢ ( r ) .

The proof of Theorem 1.2 is based on three main ingredients: a lower bound for Seshadri constants of intersections as described in [16, Section 5.4], an inequality that may be viewed as a generalization of Chebyshev’s inequality (Lemma 3.1), and an application of Theorem 1.1 to certain auxiliary closed subschemes obtained as intersections. The applicability of our generalized Chebyshev inequality, for which we provide a complete proof due to the lack of a reference, stems from the fact that our definition of 𝑚-subgeneral position of closed subschemes, even when specialized to the case of general position, allows a closed subscheme 𝑌 of codimension 𝑟 to be repeated 𝑟 times while general position is maintained. Due to this, it is not clear how to extend our results to similar inequalities involving beta constants in place of Seshadri constants, as the former inequalities use a somewhat more restrictive notion of general position.

The structure of this paper is as follows. In Section 2, we recall basic definitions and give a standard lemma with a short proof based on our general philosophy. Subsequently, we prove Theorem 1.2 in Section 3. In Section 4, we discuss the relation of Theorem 1.2 with results of Quang and others on Diophantine approximation to divisors in subgeneral position. Finally, in Sections 5 and 6, we prove the inequalities of Nochka-type.

2 Definitions and preliminaries

We begin with a brief account of the key properties of local and global height functions, following mostly the reference [38] (see also [3]). We will then introduce the notion of subgeneral position used in this paper. Finally, we revisit a standard lemma (Lemma 2.3) by giving a very simple proof of it based on the general philosophy developed in our work.

2.1 Local height functions

Let 𝑌 be a closed subscheme of a projective variety 𝑋, both defined over a number field 𝑘. For any place 𝑣 of 𝑘, one can associate a local height function (or Weil function) λ Y , v : X ⁢ ( k ) ∖ Y → R , well-defined up to O ⁢ ( 1 ) , which gives a measure of the 𝑣-adic distance of a point to 𝑌, being large when the point is close to 𝑌. If Y = D is an effective (Cartier) divisor (which we will frequently identify with the associated closed subscheme), these height functions agree with the usual height functions associated to divisors. Local height functions satisfy the following properties: if 𝑌 and 𝑍 are two closed subschemes of 𝑋, defined over 𝑘, and 𝑣 is a place of 𝑘, then up to O ⁢ ( 1 ) ,

λ Y ∩ Z , v = min ⁡ { λ Y , v , λ Z , v } , λ Y + Z , v = λ Y , v + λ Z , v , λ Y , v ≤ λ Z , v if ⁢ Y ⊂ Z .

In particular, λ Y , v is bounded from below for all P ∈ X ⁢ ( k ) ∖ Y . If ϕ : W → X is a morphism of projective varieties with ϕ ⁢ ( W ) ⊄ Y , then up to O ⁢ ( 1 ) ,

λ Y , v ⁢ ( ϕ ⁢ ( P ) ) = λ ϕ ∗ ⁢ Y , v ⁢ ( P ) for all ⁢ P ∈ W ⁢ ( k ) ∖ ϕ ∗ ⁢ Y .

Here, Y ∩ Z , Y + Z , Y ⊂ Z , and ϕ ∗ ⁢ Y are defined in terms of the associated ideal sheaves. In particular, we emphasize that if 𝑌 corresponds to the ideal sheaf I Y , then ϕ ∗ ⁢ Y is the closed subscheme corresponding to the inverse image ideal sheaf ϕ − 1 ⁢ I Y ⋅ O W .

For our purposes, it is important to note that one can set Y = Z and obtain

λ 2 ⁢ Y , v = λ Y + Y , v = λ Y , v + λ Y , v = 2 ⁢ λ Y , v ,

where 2 ⁢ Y is the closed subscheme corresponding to the ideal sheaf I Y 2 . Moreover, it holds that if Y 1 , … , Y m are closed subschemes of 𝑋 and c 1 , … , c m are positive integers, then

(2.1) λ c 1 ⁢ Y 1 ∩ ⋯ ∩ c m ⁢ Y m , v = min ⁡ { c 1 ⁢ λ Y 1 , v , … , c m ⁢ λ Y m , v } ,

λ c 1 ⁢ Y 1 + ⋯ + c m ⁢ Y m , v = c 1 ⁢ λ Y 1 , v + ⋯ + c m ⁢ λ Y m , v .

2.2 Global height functions

If 𝑌 is a closed subscheme of a projective variety 𝑋, both defined over a number field 𝑘, then a global height function h Y : X ⁢ ( k ) ∖ Y → R can be associated as follows. Let M k denote the set of all places of 𝑘, and set

h Y ⁢ ( P ) = ∑ v ∈ M k λ Y , v ⁢ ( P ) .

A subtle point here is that the definition of the λ Y , v , v ∈ M k , can be made so that ( λ Y , v ) v ∈ M k is well-defined up to a so-called M k -constant, which is a collection of real numbers ( c v ) v ∈ M K such that c v = 0 for all but finitely many 𝑣. Therefore, h Y is well-defined up to O ⁢ ( 1 ) .

For a given finite set 𝑆 of places of 𝑘, it is customary to split the above sum into the proximity function

m Y , S ⁢ ( P ) = ∑ v ∈ S λ Y , v ⁢ ( P )

and the counting function

N Y , S ⁢ ( P ) = ∑ v ∈ M k ∖ S λ Y , v ⁢ ( P ) .

Global height functions satisfy properties analogous to those stated above for local height functions, except the first property above, which becomes

h Y ∩ Z ≤ min ⁡ { h Y , h Z } + O ⁢ ( 1 ) .

When not addressed explicitly and even when any mention of M k -constants or O ⁢ ( 1 ) is omitted for brevity’s sake, all equalities and inequalities involving height functions are to be interpreted with their mild ill-definedness in mind, which is standard procedure.

A special role is played by global height functions of Cartier divisors, in particular, of ample Cartier divisors. Namely, if 𝐴 is an ample Cartier divisor, the global height function h A measures the arithmetic complexity of the point 𝑃, explaining the presence of h A on the right-hand side of the inequalities in Theorems 1.1 and 1.2. Furthermore, for an arbitrary closed subscheme 𝑌, the height function h A dominates h Y in the sense that the domain of definition of h A can be extended to all of X ⁢ ( k ) and there exists a constant 𝑐 such that

(2.2) h Y ⁢ ( P ) ≤ c ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ Y . For a proof of this statement, see [38, Proposition 4.2] (or Lemmas 2.3 and 2.4 for refined statements). As explained in this reference, the intuition for this result is that if 𝑃 is 𝑣-adically close to 𝑌 (but not on 𝑌), then 𝑃 must be 𝑣-adically complicated in terms of its coordinates after a projective embedding by a very ample multiple of the divisor 𝐴.

A further property of the local height functions ( λ Y , v ) v ∈ M k is that they are in fact bounded from below by an M k -constant, which immediately yields that inequality (2.2) can be stated for any individual λ Y , v , i.e.,

(2.3) λ Y , v ⁢ ( P ) ≤ c ⁢ h A ⁢ ( P ) .

2.3 Subgeneral position and Seshadri constants

We recall the definition of being in 𝑚-subgeneral position and the definition of Seshadri constants, both in the context of closed subschemes.

Definition 2.1

If 𝑋 is a projective variety of dimension 𝑛, we say that closed subschemes Y 1 , … , Y q of 𝑋 are in 𝑚-subgeneral position if, for every subset I ⊂ { 1 , … , q } with | I | ≤ m + 1 , we have codim ⁢ ⋂ i ∈ I Y i ≥ | I | + n − m , where we use the convention that dim ∅ = − 1 . In the case m = n , we say that the closed subschemes are in general position.

It should be noted that, even when m = n , a closed subscheme of codimension 𝑟 may be repeated 𝑟 times while maintaining general position according to our definition. This phenomenon will be important in the application of the generalized Chebyshev inequality in Section 5. We use a weighted notion of 𝑚-subgeneral position in Section 6.

We now recall the notion of a Seshadri constant for a closed subscheme relative to a nef Cartier divisor (see [14, Section 2] for further details).

Definition 2.2

Let 𝑌 be a closed subscheme of a projective variety 𝑋 and let π : X ̃ → X be the blowing-up of 𝑋 along 𝑌. Let 𝐴 be a nef Cartier divisor on 𝑋. We define the Seshadri constant ϵ Y ⁢ ( A ) of 𝑌 with respect to 𝐴 to be the real number

ϵ Y ⁢ ( A ) = sup { γ ∈ Q ≥ 0 ∣ π ∗ ⁢ A − γ ⁢ E ⁢ is ⁢ Q ⁢ -nef } ,

where 𝐸 is an effective Cartier divisor on X ̃ whose associated invertible sheaf is the dual of π − 1 ⁢ I Y ⋅ O X ̃ .

2.4 A standard lemma

We will use the following lemma (well known in the numerically equivalent case) and take this opportunity to provide a very short proof of the lemma based on the philosophy expressed in our earlier work [14].

Lemma 2.3

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘. Let A , B be effective Cartier divisors defined over 𝑘, with 𝐴 ample. Let ϵ B ⁢ ( A ) be the associated Seshadri constant. For all ϵ > 0 , there is a constant c ϵ such that

ϵ B ⁢ ( A ) ⁢ h B ≤ ( 1 + ϵ ) ⁢ h A + c ϵ .

In particular, if 𝐴 and 𝐵 are numerically equivalent ample Cartier divisors, then for all ϵ > 0 , there is a constant c ϵ such that

( 1 − ϵ ) ⁢ h A − c ϵ ≤ h B ≤ ( 1 + ϵ ) ⁢ h A + c ϵ .

Proof

Without loss of generality, we can restrict ourselves to 𝜖 such that 1 + ϵ ϵ B ⁢ ( A ) is rational. Then 1 + ϵ ϵ B ⁢ ( A ) ⁢ A − B is an ample ℚ-divisor. Let 𝑁 be a sufficiently divisible integer such that

N ⁢ ( 1 + ϵ ϵ B ⁢ ( A ) ⁢ A − B )

is an ample integral divisor. By the boundedness of heights from below, we can infer

h N ⁢ ( 1 + ϵ ϵ B ⁢ ( A ) ⁢ A − B ) ≥ − c ϵ ′

and thus

N ⁢ h B ≤ N ⁢ 1 + ϵ ϵ B ⁢ ( A ) ⁢ h A + c ϵ ′ ,

which yields the inequality. In particular, if 𝐴 and 𝐵 are numerically equivalent ample divisors, then ϵ B ⁢ ( A ) = ϵ A ⁢ ( B ) = 1 , and the result follows by applying the inequality twice, once with the roles of 𝐴 and 𝐵 switched. ∎

More generally, one easily extends this result to closed subschemes.

Lemma 2.4

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘. Let 𝐴 be an ample divisor on 𝑋 and 𝑌 a closed subscheme of 𝑋, both defined over 𝑘. Let ϵ Y ⁢ ( A ) be the associated Seshadri constant. For all ϵ > 0 , there is a constant c ϵ such that

ϵ Y ⁢ ( A ) ⁢ h Y ⁢ ( P ) ≤ ( 1 + ϵ ) ⁢ h A ⁢ ( P ) + c ϵ

for all P ∈ X ⁢ ( k ) ∖ Supp ⁡ Y .

Proof

Let π : X ̃ → X be the blowing-up of 𝑋 along 𝑌, and let 𝐸 be the associated exceptional divisor as in Definition 2.2. Then, from the definitions, ϵ Y ⁢ ( A ) = ϵ E ⁢ ( π ∗ ⁢ A ) . While π ∗ ⁢ A is not necessarily ample, it is known [12, Exercise II.7.14 (b)] that π ∗ ⁢ A − δ ⁢ E is an ample ℚ-divisor for all sufficiently small (rational) 𝛿. Then, using functoriality of heights, essentially the same proof as in Lemma 2.3 (applied to π ∗ ⁢ A and 𝐸) yields the inequality. ∎

3 Proof of Theorem 1.2

For the proof of Theorem 1.2, we need the following inequality, which is a generalization of Chebyshev’s inequality (see also work of Jensen [20, p. 245]). For lack of a reference, we provide a detailed proof. In an earlier version of this paper available from the arXiv, the proof was conducted by induction and a division into certain cases. In the present version, we instead reproduce a shorter and more elegant proof kindly provided to us by Paul Vojta.

Lemma 3.1

Let a 1 ≥ a 2 ≥ ⋯ ≥ a n ≥ 0 and b 1 , … , b n , c 1 , … , c n be nonnegative real numbers. Assume that there exists at least one index 𝑗 such that c j ≠ 0 , and let j 0 be the smallest such index. Then

(3.1) ∑ i = 1 n a i ⁢ b i ≥ ( min j 0 ≤ j ≤ n ⁡ ∑ i = 1 j b i ∑ i = 1 j c i ) ⁢ ∑ i = 1 n a i ⁢ c i .

Remark 3.2

If additionally b 1 ≥ b 2 ≥ ⋯ ≥ b n ≥ 0 and c 1 = ⋯ = c n = 1 , then the minimum on the right-hand side of (3.1) occurs at j = n and one obtains Chebyshev’s inequality

∑ i = 1 n a i ⁢ b i ≥ 1 n ⁢ ∑ i = 1 n a i ⁢ ∑ i = 1 n b i .

Similarly, if one assumes the c i are all positive and b 1 c 1 ≥ ⋯ ≥ b n c n , then the minimum on the right-hand side of (3.1) again occurs at j = n , and one obtains an inequality of Jensen [20, p. 245].

Proof

The statement is equivalent to showing that there is some j ≥ j 0 such that

L j : = ( ∑ i = 1 j c i ) ∑ i = 1 n a i b i ≥ ( ∑ i = 1 j b i ) ∑ i = 1 n a i c i = : R j .

Let a n + 1 = 0 , and note that

∑ j = 1 n ( ( a j − a j + 1 ) ⁢ ∑ i = 1 j c i ) = ∑ i = 1 n a i ⁢ c i .

Indeed, this is an application of summation by parts (or note that, for all 𝑖, the coefficient of c i on the left-hand side equals ∑ j = i n ( a j − a j + 1 ) = a i − a n + 1 = a i ). Therefore,

∑ j = 1 n ( a j − a j + 1 ) ⁢ L j = ( ∑ i = 1 n a i ⁢ c i ) ⁢ ( ∑ i = 1 n a i ⁢ b i ) .

By interchanging the roles of the b i and c i , we similarly obtain

∑ j = 1 n ( a j − a j + 1 ) ⁢ R j = ( ∑ i = 1 n a i ⁢ b i ) ⁢ ( ∑ i = 1 n a i ⁢ c i ) .

Thus,

∑ j = 1 n ( a j − a j + 1 ) ⁢ L j = ∑ j = 1 n ( a j − a j + 1 ) ⁢ R j .

Since L j = 0 and ( a j − a j + 1 ) ⁢ R j ≥ 0 for all j < j 0 , we have

∑ j = j 0 n ( a j − a j + 1 ) ⁢ L j ≥ ∑ j = j 0 n ( a j − a j + 1 ) ⁢ R j .

Finally, omitting those values of 𝑗 for which a j = a j + 1 , we obtain

∑ j = j 0 , … , n a j > a j + 1 ( a j − a j + 1 ) ⁢ ( L j − R j ) ≥ 0 .

Therefore, we conclude that L j ≥ R j for some j ≥ j 0 , unless this is the empty sum. But that can only occur if a j 0 = ⋯ = a n = 0 , which implies ∑ i = 1 n a i ⁢ c i = 0 , in which case the lemma is trivial. ∎

An immediate corollary of Lemma 3.1 is the following.

Corollary 3.3

Let a 1 ≥ a 2 ≥ ⋯ ≥ a n ≥ 0 and b 1 , … , b n , c 1 , … , c n be nonnegative real numbers. Assume that b 1 ≠ 0 . Then

( max 1 ≤ j ≤ n ⁡ ∑ i = 1 j c i ∑ i = 1 j b i ) ⁢ ∑ i = 1 n a i ⁢ b i ≥ ∑ i = 1 n a i ⁢ c i .

We now conduct the proof of Theorem 1.2.

Proof of Theorem 1.2

Let v ∈ S . In order to handle the technical difficulty that the Seshadri constants ϵ Y 1 , v ⁢ ( A ) , … , ϵ Y q , v ⁢ ( A ) may be unrelated distinct positive real numbers, we “normalize” them in a two step process. First, let τ 1 , v , … , τ q , v be real numbers in the interval [ 0 , 1 ) so that

( 1 + τ 1 , v ) ⁢ ϵ Y 1 , v ⁢ ( A ) , … , ( 1 + τ q , v ) ⁢ ϵ Y q , v ⁢ ( A )

are rational numbers. We choose τ i , v = 0 if ϵ Y i , v ⁢ ( A ) is already a rational number. For a sufficiently divisible integer c v (independent of 𝑖) such that

c v ⁢ ( 1 + τ 1 , v ) ⁢ ϵ Y 1 , v ⁢ ( A ) , … , c v ⁢ ( 1 + τ q , v ) ⁢ ϵ Y q , v ⁢ ( A )

are all integers, we set

Y ̃ i , v : = c v ( 1 + τ i , v ) ϵ Y i , v ( A ) Y i , v ,

where the multiplication of the scheme Y i , v with the integer c v ⁢ ( 1 + τ i , v ) ⁢ ϵ Y i , v ⁢ ( A ) is supposed to be understood in the sense of schemes. Observe that

(3.2) ϵ Y ̃ i , v ⁢ ( A ) = 1 c v ⁢ ( 1 + τ i , v )

for i = 1 , … , q , and therefore

1 c v ≤ 2 ⁢ ϵ Y ̃ i , v ⁢ ( A )

for i = 1 , … , q . Consequently,

1 c v ≤ 2 ⁢ min i ⁡ ϵ Y ̃ i , v ⁢ ( A ) .

Note that

(3.3) | ϵ Y ̃ i , v ⁢ ( A ) − ϵ Y ̃ i ′ , v ⁢ ( A ) | = | 1 c v ⁢ ( 1 + τ i , v ) − 1 c v ⁢ ( 1 + τ i ′ , v ) | = 1 c v ⁢ | τ i ′ , v − τ i , v ( 1 + τ i ′ , v ) ⁢ ( 1 + τ i , v ) | ≤ max k ⁡ τ k , v c v ≤ max k ⁡ τ k , v ⋅ 2 ⁢ min l ⁡ ϵ Y ̃ l , v ⁢ ( A )

is satisfied for all i , i ′ = 1 , … , q .

We introduce the following notation for intersections of the normalized subschemes:

Y ̃ I , v = ⋂ i ∈ I Y ̃ i , v .

Fix a point 𝑃 outside of the support of the given closed subschemes. Let

{ i 1 , v , … , i q , v } = { 1 , … , q }

be such that (choosing, as we may, our local height functions to be nonnegative)

(3.4) λ Y ̃ i 1 , v , v , v ⁢ ( P ) ≥ ⋯ ≥ λ Y ̃ i q , v , v , v ⁢ ( P ) ≥ 0 .

We set I j , v = { i 1 , v , … , i j , v } . Let m v be the largest index such that

⋂ j = 1 m v Y ̃ i j , v , v ≠ ∅ .

Due to the chain of inequalities (3.4) and property (2.1), we have

λ Y ̃ i j , v ⁢ v , v ⁢ ( P ) = min j ′ ≤ j ⁡ λ Y ̃ i j ′ , v , v , v ⁢ ( P ) = λ Y ̃ I j , v , v , v ⁢ ( P ) .

In particular, for j > m v , λ Y ̃ i j , v , v , v ⁢ ( P ) is bounded (by a constant independent of 𝑃) and

∑ v ∈ S ∑ i = 1 q c i , v ⁢ ϵ Y ̃ i , v ⁢ ( A ) ⁢ λ Y ̃ i , v , v ⁢ ( P ) ≤ ∑ v ∈ S ∑ j = 1 m v c i j , v ⁢ ϵ Y ̃ i j , v , v ⁢ ( A ) ⁢ λ Y ̃ I j , v , v , v ⁢ ( P ) .

Let b j , v = codim ⁡ Y ̃ I j , v , v and set b 0 , v = 0 . Let ϵ v ⁢ ( A ) = max i ⁡ ϵ Y ̃ i , v ⁢ ( A ) . We can therefore apply Corollary 3.3 (for each v ∈ S ) with

a 1 = ϵ v ⁢ ( A ) ⁢ λ Y ̃ I 1 , v , v , v ⁢ ( P ) ≥ ⋯ ≥ a m v = ϵ v ⁢ ( A ) ⁢ λ Y ̃ I m v , v , v , v ⁢ ( P ) ≥ 0 , b 1 = b 1 , v − b 0 , v , … , b m v = b m v , v − b m v − 1 , v , c j = c i j , v

to obtain, using the telescoping property for the b i ,

(3.5) ∑ v ∈ S ( max j ⁡ ∑ j ′ = 1 j c i j ′ , v b j , v ) ⁢ ∑ s = 1 m v ( b s , v − b s − 1 , v ) ⁢ ϵ v ⁢ ( A ) ⁢ λ Y ̃ I s , v , v , v ⁢ ( P ) ≥ ∑ v ∈ S ∑ j = 1 m v c i j , v ⁢ ϵ v ⁢ ( A ) ⁢ λ Y ̃ I j , v , v , v ⁢ ( P ) .

According to [16, Example 5.4.11] (which is phrased in terms of the 𝑠-invariant, which is the reciprocal of the Seshadri constant), the Seshadri constant of an intersection is bounded below by the minimum of the Seshadri constants of the factors of the intersection. Therefore, for any j ∈ { 1 , … , m v } , we have the inequality

ϵ Y ̃ I j , v , v ⁢ ( A ) ≥ min ⁡ { ϵ Y ̃ i 1 , v ⁢ ( A ) , … , ϵ Y ̃ i j , v ⁢ ( A ) } ≥ ϵ v ⁢ ( A ) − max k ⁡ τ k , v ⋅ 2 ⁢ min l ⁡ ϵ Y ̃ l , v ⁢ ( A ) ,

where the second inequality is due to (3.3). Then (3.5) implies

∑ v ∈ S ( max j ∑ j ′ = 1 j c i j ′ , v b j , v ) ∑ s = 1 m v ( b s , v − b s − 1 , v ) ( ϵ Y ̃ I s , v , v ( A ) + max k τ k , v ⋅ 2 min l ϵ Y ̃ l , v ( A ) ) λ Y ̃ I s , v , v , v ( P ) ≥ ∑ v ∈ S ∑ j = 1 m v c i j , v ⁢ ϵ v ⁢ ( A ) ⁢ λ Y ̃ I j , v , v , v ⁢ ( P ) ≥ ∑ v ∈ S ∑ j = 1 m v c i j , v ⁢ ϵ Y ̃ i j , v , v ⁢ ( A ) ⁢ λ Y ̃ I j , v , v , v ⁢ ( P ) .

We now note that if we form a new list with the closed subscheme Y ̃ I s , v , v repeated b s , v − b s − 1 , v times (and omitted if b s , v − b s − 1 , v = 0 ), then the resulting closed subschemes are in general position. By Theorem 1.1, there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that

∑ v ∈ S ∑ s = 1 m v ( b s , v − b s − 1 , v ) ⁢ ϵ Y ̃ I s , v , v ⁢ ( A ) ⁢ λ Y ̃ I s , v , v , v ⁢ ( P ) < ( n + 1 + ϵ ′ ) ⁢ h A ⁢ ( P )

for all points P ∈ X ⁢ ( k ) ∖ Z . We also observe that, by (3.2) and additivity for local heights, for i = 1 , … , q , we have

ϵ Y ̃ i , v ⁢ ( A ) ⁢ λ Y ̃ i , v , v ⁢ ( P ) = 1 c v ⁢ ( 1 + τ i , v ) ⋅ c v ⁢ ( 1 + τ i , v ) ⁢ ϵ Y i , v ⁢ ( A ) ⁢ λ Y i , v , v ⁢ ( P ) = ϵ Y i , v ⁢ ( A ) ⁢ λ Y i , v , v ⁢ ( P ) .

For given ϵ ′′ > 0 , by choosing the quantities τ k , v sufficiently small (depending only on data involving the Y i , v , including the finite number of pertaining domination inequalities of the form (2.3) for the Y i , v ), we have

∑ v ∈ S ∑ s = 1 m v ( b s , v − b s − 1 , v ) ⁢ ( max k ⁡ τ k , v ⋅ 2 ⁢ min l ⁡ ϵ Y ̃ l , v ⁢ ( A ) ) ⁢ λ Y ̃ I s , v , v , v ⁢ ( P ) ≤ ∑ v ∈ S ∑ s = 1 m v ( b s , v − b s − 1 , v ) ⁢ ( max k ⁡ τ k , v ⋅ 2 ⁢ ϵ Y ̃ i s , v , v ⁢ ( A ) ) ⁢ λ Y ̃ i s , v , v , v ⁢ ( P ) ≤ ∑ v ∈ S ∑ s = 1 m v ( b s , v − b s − 1 , v ) ⁢ ( max k ⁡ τ k , v ⋅ 2 ⁢ ϵ Y i s , v , v ⁢ ( A ) ) ⁢ λ Y i s , v , v , v ⁢ ( P ) < ϵ ′′ ⁢ h A ⁢ ( P ) .

Taking W = Supp ⁡ Y ̃ I j , v , v ≠ ∅ , we also find

∑ j ′ = 1 j c i j ′ , v b j , v ≤ α v ⁢ ( W ) codim ⁡ W .

Note that, although the indices i 1 , v , … , i m v , v , v ∈ S , depended on the point 𝑃, there are only finitely many possibilities for these indices (and so only finitely many applications of Theorem 1.1 are needed). Putting everything together, we find that there exists a proper Zariski-closed set 𝑍 of 𝑋 such that

∑ v ∈ S ∑ i = 1 q c i , v ⁢ ϵ Y i , v ⁢ ( A ) ⁢ λ Y i , v , v ⁢ ( P ) ≤ ( ( n + 1 ) ⁢ max v ∈ S ∅ ⊊ W ⊊ X ⁡ α v ⁢ ( W ) codim ⁡ W + ϵ ) ⁢ h A ⁢ ( P )

for all points P ∈ X ⁢ ( k ) ∖ Z , as desired. ∎

4 Results of Quang and relations to other work on Schmidt’s subspace theorem for divisors in subgeneral position

Quang [26] proved the following generalization of Evertse–Ferretti’s theorem for divisors in subgeneral position.

Theorem 4.1

Theorem 4.1 (Quang)

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘. Let 𝑆 be a finite set of places of 𝑘. Let D 1 , … , D q be effective Cartier divisors on 𝑋, defined over 𝑘, in 𝑚-subgeneral position. Suppose that there exists an ample Cartier divisor 𝐴 on 𝑋 and positive integers d i such that D i ∼ d i ⁢ A for all 𝑖. Let ϵ > 0 . Then there exists a proper Zariski-closed subset Z ⊂ X such that, for all points P ∈ X ⁢ ( k ) ∖ Z ,

∑ i = 1 q m D i , S ⁢ ( P ) d i < ( ( m − n + 1 ) ⁢ ( n + 1 ) + ϵ ) ⁢ h A ⁢ ( P ) .

This improved an earlier result of the second author [18] which had a factor of

m ⁢ ( m − 1 ) ⁢ ( n + 1 ) m + n − 2 + ϵ

on the right-hand side.

Quang’s theorem was refined by Ji, Yan, and Yu in [15] using the notion of the index.

Definition 4.2

If 𝑋 is a projective variety of dimension 𝑛, we say that closed subschemes Y 1 , … , Y q of 𝑋 are in 𝑚-subgeneral position with index 𝜅 if Y 1 , … , Y q are in 𝑚-subgeneral position and, for every subset J ⊂ { 1 , … , q } with | J | ≤ κ , we have

codim ⁢ ⋂ j ∈ J Y j ≥ | J | .

Then Ji, Yan, and Yu proved the following theorem.

Theorem 4.3

Theorem 4.3 (Ji, Yan, Yu)

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘. Let 𝑆 be a finite set of places of 𝑘. Let D 1 , … , D q be effective Cartier divisors on 𝑋, defined over 𝑘, in 𝑚-subgeneral position with index 𝜅. Suppose that there exists an ample Cartier divisor 𝐴 on 𝑋 and positive integers d i such that D i ∼ d i ⁢ A for all 𝑖. Let ϵ > 0 . Then there exists a proper Zariski-closed subset Z ⊂ X such that, for all points P ∈ X ⁢ ( k ) ∖ Z ,

∑ i = 1 q m D i , S ⁢ ( P ) d i < ( ( m − n max ⁡ { 1 , min ⁡ { m − n , κ } } + 1 ) ⁢ ( n + 1 ) + ϵ ) ⁢ h A ⁢ ( P ) .

Remark 4.4

Ji, Yan, and Yu also claim that one can prove an inequality with

max ⁡ { m 2 ⁢ κ , 1 } ⁢ ( n + 1 )

on the right-hand side [15, Theorems 1.1 and 5.1]. As they note, when m ≤ 2 ⁢ κ , this gives a (claimed) coefficient of n + 1 + ϵ on the right-hand side. As observed already by Quang in [27], their proof of this claim is incorrect, and we take the opportunity to give an explicit counterexample. Fix 𝑘 and 𝑆 with | S | > 1 . Take, for instance, 5 distinct lines L 1 , … , L 5 in the plane with 4 of them passing through a point 𝑄 and let D = L 1 + ⋯ + L 5 . Then m = 4 and κ = 2 . Any line 𝐿 passing through 𝑄 intersects the 5 lines in 2 points and so contains an infinite set R L of ( D , S ) -integral points. Then, for such a line 𝐿 and points P ∈ R L ⊂ L ⁢ ( k ) ,

∑ i = 1 5 m L i , S ⁢ ( P ) = 5 ⁢ h ⁢ ( P ) .

Since such lines 𝐿 are Zariski dense in P 2 , the best coefficient one can have in this case is 5 (and not 3 as given in their claim). More generally, it is easy to see that if one has at least m + 1 hyperplanes in P n in 𝑚-subgeneral position (and also not in ( m − 1 ) -subgeneral position), one cannot possibly do better than a coefficient of m + 1 on the right-hand side, regardless of the index 𝜅.

Shi [37] improved Theorem 4.3 in the Nevanlinna theory setting. The analogue of Shi’s result in Diophantine approximation replaces the quantity

( m − n max ⁡ { 1 , min ⁡ { m − n , κ } } + 1 ) ⁢ ( n + 1 )

in Theorem 4.3 by ( m − n κ + 1 ) ⁢ ( n + 1 ) .

More recently, Quang independently introduced the notion of a distributive constant Δ for a family of divisors in [27], and in the recent preprint [28], Quang extended this notion to closed subschemes^[1].

Definition 4.5

The distributive constant of a family of closed subschemes

Y = { Y 1 , … , Y q }

on a projective variety 𝑋 is given by

δ Y = max ∅ ≠ J ⊂ { 1 , … , q } ⁡ max ⁡ { 1 , # ⁢ J codim ⁢ ⋂ j ∈ J Supp ⁡ Y j } ,

where we set codim ⁡ ∅ = ∞ .

Then, in [28], Quang proves the following theorem.

Theorem 4.6

Theorem 4.6 (Quang [28])

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘, and let 𝑆 be a finite set of places of 𝑘. For each v ∈ S , let Y 1 , v , … , Y q , v be closed subschemes of 𝑋, defined over 𝑘. Let Δ = max v ∈ S ⁡ δ Y v be the maximum of the distributive constants of Y v = { Y 1 , v , … , Y q , v } . Let 𝐴 be an ample Cartier divisor on 𝑋, and ϵ > 0 . Then there exists a proper Zariski-closed set 𝑍 of 𝑋 such that

∑ v ∈ S ∑ i = 1 q ϵ Y i , v ⁢ ( A ) ⁢ λ Y i , v , v ⁢ ( P ) < ( Δ ⁢ ( n + 1 ) + ϵ ) ⁢ h A ⁢ ( P )

for all points P ∈ X ⁢ ( k ) ∖ Z .

As noted in [27], if the Y i , v are taken to be effective divisors in 𝑚-subgeneral position, then

Δ ≤ m − n + 1 ,

and if the Y i , v are divisors in 𝑚-subgeneral position with index 𝜅, then

Δ ≤ m − n κ + 1 .

Thus, Theorem 4.6 contains the previous results of Quang, Ji–Yan–Yu, and Shi. If we set c i , v = 1 for all 𝑖 and 𝑣, then in comparison with Theorem 1.2,

δ Y v = max ⁡ { 1 , max ∅ ⊊ W ⊊ X ⁡ α v ⁢ ( W ) codim ⁡ W } .

This follows from the definitions and the easy observation that, in the maximum over 𝑊, we may restrict to the case where 𝑊 is the intersection of a subset of closed subschemes of Y v .

Thus, we may view Theorem 1.2 as a weighted version of Quang’s Theorem 4.6, with an improvement in the coefficient of the height in Theorem 1.2 in certain cases (due to the maximum with 1 in Quang’s definition of δ Y ) .

In particular, Theorem 1.2 gives weighted generalizations of the results of Quang [26], Ji–Yan–Yu [15], and Shi [37], in the general context of closed subschemes, using weighted generalizations of the notions of 𝑚-subgeneral position (see Section 6) and the index.

Note that Quang’s proof of Theorem 4.6 is an adaptation of our proof of Theorem 1.1, while our proof of Theorem 1.2 is derived as a formal consequence of the statement of Theorem 1.1 using properties of heights, Seshadri constants, and the generalized Chebyshev inequality of Lemma 3.1.

5 A general inequality of Nochka-type

This section is influenced by the approach of Vojta in [42] to the seminal work of Nochka on weights [24] (see Remark 5.3). Let 𝑋 be a projective variety of dimension 𝑛 and let D 1 , … , D q be effective Cartier divisors on 𝑋, everything defined over a number field 𝑘. We first restrict to the classical case where c i = 1 for i = 1 , … , q . Then, for a closed subset W ⊂ X , we let

α ⁢ ( W ) = # ⁢ { i ∣ W ⊂ Supp ⁡ D i } .

We prove the following easy corollary to Theorem 1.2.

Corollary 5.1

Let W 0 be a closed subset of 𝑋. Let 𝐴 be an ample Cartier divisor on 𝑋, and ϵ > 0 . Then there exists a proper Zariski-closed set 𝑍 of 𝑋 such that

∑ i = 1 q ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) < ( α ⁢ ( W 0 ) + ( n + 1 ) ⁢ max ∅ ⊊ W ⊊ X ⁡ α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W + ϵ ) ⁢ h A ⁢ ( P )

for all points P ∈ X ⁢ ( k ) ∖ Z .

Proof

After reindexing, suppose that W 0 ⊂ Supp ⁡ D j for the α ⁢ ( W 0 ) values of 𝑗 satisfying q − α ⁢ ( W 0 ) + 1 ≤ j ≤ q . Let q ′ = q − α ⁢ ( W 0 ) . Let

α ′ ⁢ ( W ) = # ⁢ { i ≤ q ′ ∣ W ⊂ Supp ⁡ D i } .

Then, clearly, α ′ ⁢ ( W ) = α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) . Using Lemma 2.3, we have the inequality

∑ i = q ′ + 1 q ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) ≤ ( α ⁢ ( W 0 ) + ϵ ) ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ ⋃ i = q ′ + 1 q D i .

Then, using the above and Theorem 1.2, there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that

∑ i = 1 q ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) = ∑ i = 1 q ′ ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) + ∑ i = q ′ + 1 q ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P )

≤ ( ( n + 1 ) ⁢ max ∅ ⊊ W ⊊ X ⁡ α ′ ⁢ ( W ) codim ⁡ W + ϵ ) ⁢ h A ⁢ ( P ) + ( α ⁢ ( W 0 ) + ϵ ) ⁢ h A ⁢ ( P )

≤ ( α ⁢ ( W 0 ) + ( n + 1 ) ⁢ max ∅ ⊊ W ⊊ X ⁡ α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W + 2 ⁢ ϵ ) ⁢ h A ⁢ ( P )

for all points P ∈ X ⁢ ( k ) ∖ Z . ∎

Building on the method of proof of [42, Main Theorem], we now prove Theorem 1.6, which we restate for convenience. Recall that D I = ⋂ i ∈ I D i .

Theorem 5.2

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘 and let 𝑆 be a finite set of places of 𝑘. Let D 1 , … , D q be effective Cartier divisors on 𝑋, defined over 𝑘, in 𝑚-subgeneral position. We assume the following Bezout property holds for intersections among the divisors: if I , J ⊂ { 1 , … , q } , then

codim ⁡ D I ∪ J = codim ⁡ ( D I ∩ D J ) ≤ codim ⁡ D I + codim ⁡ D J .

Let 𝐴 be an ample divisor on 𝑋 and let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that, for all P ∈ X ⁢ ( k ) ∖ Z ,

∑ i = 1 q ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) < ( 3 2 ⁢ ( 2 ⁢ m − n + 1 ) + ϵ ) ⁢ h A ⁢ ( P ) .

Remark 5.3

We briefly describe the main idea of the proof. Vojta constructs the classical Nochka weights (for hyperplanes) as slopes of line segments connecting the lower convex hull of the set of points consisting of ( α ⁢ ( W ) , codim ⁡ W ) , ∅ ⊊ W ⊂ X , and the point P = ( 2 ⁢ m − n + 1 , n + 1 ) . When this lower convex hull consists of just ( 0 , 0 ) and 𝑃, the theorem follows easily from Theorem 1.2. Otherwise, we focus on the last line segment associated to the lower convex hull in the “Nochka diagram,” connecting (for some W 0 ) the points ( α ⁢ ( W 0 ) , codim ⁡ W 0 ) and 𝑃, with slope 𝜎. Then we (essentially) assign the weight 0 to divisors D i with W 0 ⊂ Supp ⁡ D i , and the weight 𝜎 to all other divisors D i . This partial use of the Nochka diagram loses a factor of 3 2 (compared to the case of hyperplanes), but works in more generality.

Proof

Suppose first that, for every W ⊂ X , W ≠ ∅ ,

codim ⁡ W ≥ n + 1 2 ⁢ m − n + 1 ⁢ α ⁢ ( W ) .

Then it follows immediately from Theorem 1.2 that (1.2) holds with the better coefficient 2 ⁢ m − n + 1 + ϵ on the right-hand side.

Otherwise, we choose W 0 ≠ ∅ in Corollary 5.1 such that the quantity

n + 1 − codim ⁡ W 2 ⁢ m − n + 1 − α ⁢ ( W )

is maximized at W = W 0 . Replacing W 0 by the intersection of the divisors containing W 0 , we may assume that W 0 is an intersection D I for some I ⊂ { 1 , … , q } . Let

σ = n + 1 − codim ⁡ W 0 2 ⁢ m − n + 1 − α ⁢ ( W 0 )

(when the divisors D i are hyperplanes, this agrees with the slope σ = τ between the points P s and P s + 1 in the corresponding Nochka diagram [42, p. 232]).

Let ∅ ⊊ W ⊊ X . By Corollary 5.1, we need to show that

α ⁢ ( W 0 ) + ( n + 1 ) ⁢ α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W ≤ 3 2 ⁢ ( 2 ⁢ m − n + 1 ) .

By replacing 𝑊 by the intersection of the divisors containing 𝑊, it suffices to consider the case that W = D J for some nonempty J ⊂ { 1 , … , q } (the case α ⁢ ( W ) = 0 , J = ∅ , follows easily from 𝑚-subgeneral position).

We first claim that

α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W ≤ 1 σ .

We consider two cases.

Case I: W ∩ W 0 = ∅ . In this case, by our Bezout assumption,

codim ⁡ W + codim ⁡ W 0 ≥ n + 1 ,

and from 𝑚-subgeneral position, we have

codim ⁡ W 0 ≥ α ⁢ ( W 0 ) + n − m , codim ⁡ W ≥ α ⁢ ( W ) + n − m .

Then

1 σ = 2 ⁢ m − n + 1 − α ⁢ ( W 0 ) n + 1 − codim ⁡ W 0 ≥ m + 1 − codim ⁡ W 0 n + 1 − codim ⁡ W 0 = 1 + m − n n + 1 − codim ⁡ W 0 ≥ 1 + m − n codim ⁡ W = codim ⁡ W + m − n codim ⁡ W ≥ α ⁢ ( W ) codim ⁡ W ≥ α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W ,

as desired.

Case II: W ∩ W 0 ≠ ∅ . We use the two inequalities

α ⁢ ( W ∪ W 0 ) + α ⁢ ( W ∩ W 0 ) ≥ α ⁢ ( W ) + α ⁢ ( W 0 )

and (from our Bezout assumption)

codim ⁡ W ≥ codim ⁡ ( W ∩ W 0 ) − codim ⁡ W 0 .

First, suppose that codim ⁡ ( W ∩ W 0 ) = codim ⁡ W 0 . Since trivially α ⁢ ( W ∩ W 0 ) ≥ α ⁢ ( W 0 ) , from the definition of W 0 , we must have α ⁢ ( W ∩ W 0 ) = α ⁢ ( W 0 ) . This implies α ⁢ ( W ∪ W 0 ) = α ⁢ ( W ) and then

α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W = 0 ≤ 1 σ .

Otherwise, if codim ⁡ ( W ∩ W 0 ) > codim ⁡ W 0 , then

α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W ≤ α ⁢ ( W ∩ W 0 ) − α ⁢ ( W 0 ) codim ⁡ ( W ∩ W 0 ) − codim ⁡ W 0 ≤ 1 σ ,

where the second inequality follows from the definition of W 0 and 𝜎. Then, in either case,

α ⁢ ( W 0 ) + ( n + 1 ) ⁢ α ⁢ ( W ) − α ⁢ ( W ∪ W 0 ) codim ⁡ W ≤ α ⁢ ( W 0 ) + n + 1 σ .

Finally, we note that, from our assumptions, P = ( α ⁢ ( W 0 ) , codim ⁡ W 0 ) lies below the line

y = n + 1 2 ⁢ m − n + 1 ⁢ x .

From 𝑚-subgeneral position, it also lies on or to the left of the line y = x + n − m . Therefore, 𝑃 must lie below and to the left of the point W = ( 2 ⁢ m − n + 1 2 , n + 1 2 ) , which is the intersection of the two given lines. Thus,

α ⁢ ( W 0 ) < 2 ⁢ m − n + 1 2 , codim ⁡ W 0 < n + 1 2 .

Since σ > n + 1 2 ⁢ m − n + 1 by assumption, it follows that

α ⁢ ( W 0 ) + n + 1 σ < 3 2 ⁢ ( 2 ⁢ m − n + 1 ) ,

as desired. ∎

Remark 5.4

From an examination of the proof, it is possible to slightly improve inequality (1.2). For instance, at the end of the proof, one could use the better estimate

α ⁢ ( W 0 ) ≤ ⌊ 2 ⁢ m − n 2 ⌋ ,

and similarly slightly improve the used lower bound for 𝜎. We state the inequality in the present form both for its simplicity and easy relation with Nochka–Ru–Wong’s inequality, and since we anticipate that further refinements of these methods may lead to more significant improvements.

6 Nochka–Ru–Wong for surfaces and threefolds

We first extend the notion of 𝑚-subgeneral position to closed subschemes Y 1 , … , Y q with nonnegative weights c 1 , … , c q (we will reserve ω i to denote certain Nochka-type weights below). Following the introduction, we write

α ⁢ ( W ) = ∑ i W ⊂ Supp ⁡ Y i c i .

Then we say that Y 1 , … , Y q with weights c 1 , … , c q are in 𝑚-subgeneral position if, for every closed subset ∅ ≠ W ⊂ X , we have

codim ⁡ W ≥ α ⁢ ( W ) + n − m .

We now prove a general (weighted) version of Nochka–Ru–Wong’s inequality for closed subschemes, assuming that one can construct “Nochka weights” that satisfy certain properties. We will subsequently apply this result with appropriately chosen weights in dimensions at most 3.

Theorem 6.1

Let 𝑋 be a projective variety of dimension 𝑛 defined over a number field 𝑘 and let 𝑆 be a finite set of places of 𝑘. Let Y 1 , … , Y q be closed subschemes of 𝑋, defined over 𝑘, with nonnegative real weights c 1 , … , c q . Suppose that there exist nonnegative real weights ω 1 , … , ω q ≥ 0 , not all 0, such that, for any nonempty closed subset W ⊂ X ,

(6.1) α Nochka ( W ) : = ∑ i W ⊂ Supp ⁡ Y i c i ω i ≤ codim W .

Let τ = max i ⁡ ω i and let

B = n + 1 τ + ∑ i = 1 q c i ⁢ ( 1 − ω i τ ) .

Let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that

∑ i = 1 q c i ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) < ( B + ϵ ) ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ Z .

Proof

We split the proximity functions as

∑ i = 1 q c i ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) = ∑ i = 1 q ( c i − c i ⁢ ω i τ ) ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) + 1 τ ⁢ ∑ i = 1 q c i ⁢ ω i ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) .

We have the following trivial bound coming from Lemma 2.4:

∑ i = 1 q ( c i − c i ⁢ ω i τ ) ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) ≤ ∑ i = 1 q ( c i − c i ⁢ ω i τ + ϵ ) ⁢ h A ⁢ ( P ) .

On the other hand, by Theorem 1.2 applied with α Nochka ⁢ ( W ) from (6.1), there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that

1 τ ⁢ ∑ i = 1 q c i ⁢ ω i ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) ≤ ( n + 1 τ + ϵ ) ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ Z . Combining the above inequalities yields

∑ i = 1 q c i ⁢ ϵ Y i ⁢ ( A ) ⁢ m Y i , S ⁢ ( P ) < ( B + ϵ ) ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ Z , as desired. ∎

We now show that, under mild hypotheses, one can find suitable “Nochka weights” in dimensions at most 3 and derive a generalization of Nochka–Ru–Wong’s inequalities.

Remark 6.2

A key feature in dimensions at most 3 is that the lower convex hull in the “Nochka diagram” (see Remark 5.3) yields at most 2 line segments, which permits a finer analysis of the problem.

Theorem 6.3

Let 𝑋 be a projective variety of dimension n ≤ 3 defined over a number field 𝑘 and let 𝑆 be a finite set of places of 𝑘. Let D 1 , … , D q be ample effective Cartier divisors on 𝑋, defined over 𝑘, with nonnegative real weights c 1 , … , c q , in 𝑚-subgeneral position. Suppose that Supp ⁡ D i is irreducible for all 𝑖. Let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that

∑ i = 1 q c i ⁢ ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) < ( 2 ⁢ m − n + 1 + ϵ ) ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ Z .

Proof

When n = 1 , this is immediate from the definition of 𝑚-subgeneral position and Theorem 1.2. Suppose n ∈ { 2 , 3 } . Let

c = max W ⊂ X codim ⁡ W = 1 ⁡ α ⁢ ( W ) = max W ⊂ X codim ⁡ W = 1 ⁡ α ⁢ ( W ) codim ⁡ W .

First, we note that, by Theorem 1.2, the desired result holds if

α ⁢ ( W ) codim ⁡ W ≤ 2 ⁢ m − n + 1 n + 1

for all nonempty proper closed subsets W ⊂ X . By definition of 𝑚-subgeneral position,

α ⁢ ( W ) ≤ codim ⁡ W + m − n ,

and if codim ⁡ W ≥ n + 1 2 , then

α ⁢ ( W ) codim ⁡ W ≤ 1 + m − n codim ⁡ W ≤ 1 + 2 ⁢ ( m − n ) n + 1 = 2 ⁢ m − n + 1 n + 1 .

For n ∈ { 2 , 3 } , codim ⁡ W ≥ n + 1 2 if and only if codim ⁡ W > 1 . Thus, the result follows from Theorem 1.2 unless

(6.2) c > 2 ⁢ m − n + 1 n + 1 ,

which we now assume.

Let W 0 be an irreducible closed subset of 𝑋 with codim ⁡ W 0 = 1 such that c = α ⁢ ( W 0 ) . We define

ω i = { 1 c if Supp ⁡ D i = W 0 n 2 ⁢ m − n + 1 − c otherwise .

Condition (6.2) implies that 1 c < n 2 ⁢ m − n + 1 − c and so in particular ω i ≤ n 2 ⁢ m − n + 1 − c for all 𝑖.

We claim that, for any nonempty closed subset W ⊂ X ,

∑ i W ⊂ Supp ⁡ D i c i ⁢ ω i ≤ codim ⁡ W .

Suppose first that dim W = 0 . From 𝑚-subgeneral position, c ≤ 1 + m − n and α ⁢ ( W ) ≤ m . Then

∑ i W ⊂ Supp ⁡ D i c i ⁢ ω i ≤ n 2 ⁢ m − n + 1 − c ⁢ α ⁢ ( W ) ≤ n m ⁢ m ≤ n = codim ⁡ W .

Suppose now that codim ⁡ W = 1 . We may assume 𝑊 is irreducible and W = D j for some 𝑗. If W = W 0 , then from the definitions, we have

∑ i W ⊂ Supp ⁡ D i c i ⁢ ω i = 1 c ⁢ α ⁢ ( W 0 ) = 1 .

Otherwise, since the divisors D i are ample, we have codim ⁡ ( W ∩ W 0 ) = 2 and

α ⁢ ( W ) + c = α ⁢ ( W ) + α ⁢ ( W 0 ) ≤ α ⁢ ( W ∩ W 0 ) ≤ codim ⁡ ( W ∩ W 0 ) + m − n = m − n + 2 .

Then

∑ i W ⊂ Supp ⁡ D i c i ⁢ ω i = n 2 ⁢ m − n + 1 − c ⁢ α ⁢ ( W ) ≤ n ⁢ ( m − n + 2 − c ) 2 ⁢ m − n + 1 − c .

Suppose first that n = 2 . Then, from 𝑚-subgeneral position with W = X , we have m ≥ n ≥ 2 . Then, from (6.2), c ≥ 2 ⁢ m − 1 3 ≥ 1 , and this implies

n ⁢ ( m − n + 2 − c ) 2 ⁢ m − n + 1 − c = 2 ⁢ ( m − c ) 2 ⁢ m − 1 − c ≤ 1 = codim ⁡ W ,

as desired. If n = 3 , by (6.2), c > 2 ⁢ m − 2 4 = m − 1 2 , and this implies

(6.3) n ⁢ ( m − n + 2 − c ) 2 ⁢ m − n + 1 − c = 3 ⁢ ( m − 1 − c ) 2 ⁢ m − 2 − c ≤ 1 = codim ⁡ W .

Finally, we consider the case n = 3 and dim W = 1 , codim ⁡ W = 2 . Suppose first that W ⊂ W 0 . Let α ′ = α ⁢ ( W ) − α ⁢ ( W 0 ) = α ⁢ ( W ) − c . Then

α ′ + c = α ⁢ ( W ) ≤ codim ⁡ ( W ) + m − n = m − 1 .

From the definitions and the calculation in (6.3), we find

∑ i W ⊂ Supp ⁡ D i c i ⁢ ω i = α ′ ⁢ n 2 ⁢ m − n + 1 − c + c ⁢ 1 c ≤ 3 ⁢ ( m − 1 − c ) 2 ⁢ m − 2 − c + 1 ≤ 2 = codim ⁡ W .

Suppose now that W ⊄ W 0 . Then, from ampleness, dim ( W ∩ W 0 ) = 0 , and from 𝑚-subgeneral position,

α ⁢ ( W ) + c = α ⁢ ( W ) + α ⁢ ( W 0 ) ≤ α ⁢ ( W ∩ W 0 ) ≤ codim ⁡ ( W ∩ W 0 ) + m − n = m .

Therefore, since m ≥ n ≥ 3 and c > m − 1 2 ≥ 1 , we easily find

∑ i W ⊂ Supp ⁡ D i c i ⁢ ω i = α ⁢ ( W ) ⁢ n 2 ⁢ m − n + 1 − c ≤ 3 ⁢ ( m − c ) 2 ⁢ m − 2 − c ≤ 2 = codim ⁡ W .

This proves (6.1) of Theorem 6.1. Finally, we have

τ = max ⁡ ω i = n 2 ⁢ m − n + 1 − c

and we compute

B = n + 1 τ + ∑ i = 1 q c i ⁢ ( 1 − ω i τ ) = n + 1 τ + ∑ i Supp ⁡ D i = W 0 c i ⁢ ( 1 − 1 c ⁢ τ ) = n + 1 τ + c ⁢ ( 1 − 1 c ⁢ τ ) = n + 1 τ + c − 1 τ = n τ + c = ( 2 ⁢ m − n + 1 − c ) + c = 2 ⁢ m − n + 1 .

Then an application of Theorem 6.1 completes the proof. ∎

Finally, under some additional hypotheses, we can remove the condition that Supp ⁡ D i is irreducible. In particular, Theorem 1.7 follows from the next theorem.

Theorem 6.4

Let 𝑋 be a nonsingular projective variety of dimension n ≤ 3 , defined over a number field 𝑘, with Picard number ρ = 1 . Let 𝑆 be a finite set of places of 𝑘. Let D 1 , … , D q be effective divisors on 𝑋, defined over 𝑘, with nonnegative weights c 1 , … , c q , in 𝑚-subgeneral position. Let 𝐴 be an ample divisor on 𝑋 and let ϵ > 0 . Then there exists a proper Zariski-closed subset 𝑍 of 𝑋 such that

∑ i = 1 q c i ⁢ ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) < ( 2 ⁢ m − n + 1 + ϵ ) ⁢ h A ⁢ ( P )

for all P ∈ X ⁢ ( k ) ∖ Z .

Proof

For i = 1 , … , q , write

D i = ∑ j = 1 r i D i ⁢ j ,

where the D i ⁢ j are (not necessarily distinct) effective divisors with Supp ⁡ D i ⁢ j irreducible. We have

m D i , S ⁢ ( P ) = ∑ j = 1 r i m D i ⁢ j , S ⁢ ( P ) .

Since ρ = 1 , we have D i ⁢ j ≡ a i ⁢ j ⁢ A for some a i ⁢ j > 0 , and ϵ D i ⁢ j ⁢ ( A ) = 1 a i ⁢ j . Similarly,

D i ≡ ∑ j = 1 r i a i ⁢ j ⁢ A ,

and so

ϵ D i ⁢ ( A ) = 1 ∑ i = 1 r i a i ⁢ j .

Then we find

∑ j = 1 r i ϵ D i ⁢ ( A ) ϵ D i ⁢ j ⁢ ( A ) = 1 .

Now we write

(6.4) ∑ i = 1 q c i ⁢ ϵ D i ⁢ ( A ) ⁢ m D i , S ⁢ ( P ) = ∑ i = 1 q ∑ j = 1 r i ( c i ⁢ ϵ D i ⁢ ( A ) ϵ D i ⁢ j ⁢ ( A ) ) ⁢ ϵ D i ⁢ j ⁢ ( A ) ⁢ m D i ⁢ j , S ⁢ ( P ) .

We claim that the divisors D i ⁢ j with weights

c i ⁢ ϵ D i ⁢ ( A ) ϵ D i ⁢ j ⁢ ( A ) , i = 1 , … , q , j = 1 , … , r i ,

are in 𝑚-subgeneral position. Let W ⊂ X be a nonempty closed subset. Then

∑ i , j W ⊂ Supp ⁡ D i ⁢ j c i ⁢ ϵ D i ⁢ ( A ) ϵ D i ⁢ j ⁢ ( A ) ≤ ∑ i W ⊂ Supp ⁡ D i c i ⁢ ∑ j = 1 r i ϵ D i ⁢ ( A ) ϵ D i ⁢ j ⁢ ( A ) = ∑ i W ⊂ Supp ⁡ D i c i .

Since the divisors D 1 , … , D q , with weights c 1 , … , c q , are in 𝑚-subgeneral position, the claim follows. We complete the proof by using (6.4) and Theorem 6.3, applied to the ample divisors D i ⁢ j (with appropriate weights). ∎

Funding source: Simons Foundation

Award Identifier / Grant number: 963755-GH

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2001205

Award Identifier / Grant number: DMS-2302298

Funding statement: The first author was supported by a grant from the Simons Foundation (Grant Number 963755-GH). The second author was supported in part by NSF grants DMS-2001205 and DMS-2302298, and a Simons Fellowship from the Simons Foundation.

Acknowledgements

We thank Paul Vojta for providing us with an elegant proof of Lemma 3.1. We thank an anonymous referee for a close reading of the paper and several helpful suggestions.

References

[1] P. Autissier, Géométries, points entiers et courbes entières, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 2, 221–239. 10.24033/asens.2094Search in Google Scholar

[2] P. Autissier, Sur la non-densité des points entiers, Duke Math. J. 158 (2011), no. 1, 13–27. 10.1215/00127094-1276292Search in Google Scholar

[3] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Math. Monogr. 4, Cambridge University, Cambridge 2006. Search in Google Scholar

[4] W. Chen, Cartan’s conjecture: Defect relations for meromorphic maps from parabolic manifold to projective space, Ph.D. Thesis, University of Notre Dame, 1987. Search in Google Scholar

[5] P. Corvaja, A. Levin and U. Zannier, Integral points on threefolds and other varieties, Tohoku Math. J. (2) 61 (2009), no. 4, 589–601. 10.2748/tmj/1264084501Search in Google Scholar

[6] P. Corvaja and U. Zannier, A subspace theorem approach to integral points on curves, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 267–271. 10.1016/S1631-073X(02)02240-9Search in Google Scholar

[7] P. Corvaja and U. Zannier, On a general Thue’s equation, Amer. J. Math. 126 (2004), no. 5, 1033–1055. 10.1353/ajm.2004.0034Search in Google Scholar

[8] P. Corvaja and U. Zannier, On integral points on surfaces, Ann. of Math. (2) 160 (2004), no. 2, 705–726. 10.4007/annals.2004.160.705Search in Google Scholar

[9] J.-H. Evertse and R. G. Ferretti, Diophantine inequalities on projective varieties, Int. Math. Res. Not. IMRN 2002 (2002), no. 25, 1295–1330. 10.1155/S107379280210804XSearch in Google Scholar

[10] J.-H. Evertse and R. G. Ferretti, A generalization of the Subspace Theorem with polynomials of higher degree, Diophantine approximation, Dev. Math. 16, Springer, Vienna (2008), 175–198. 10.1007/978-3-211-74280-8_9Search in Google Scholar

[11] R. G. Ferretti, Mumford’s degree of contact and Diophantine approximations, Compos. Math. 121 (2000), no. 3, 247–262. 10.1023/A:1001726515286Search in Google Scholar

[12] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar

[13] G. Heier and A. Levin, On the degeneracy of integral points and entire curves in the complement of nef effective divisors, J. Number Theory 217 (2020), 301–319. 10.1016/j.jnt.2020.05.013Search in Google Scholar

[14] G. Heier and A. Levin, A generalized Schmidt subspace theorem for closed subschemes, Amer. J. Math. 143 (2021), no. 1, 213–226. 10.1353/ajm.2021.0008Search in Google Scholar

[15] Q. Ji, Q. Yan and G. Yu, Holomorphic curves into algebraic varieties intersecting divisors in subgeneral position, Math. Ann. 373 (2019), no. 3–4, 1457–1483. 10.1007/s00208-018-1661-4Search in Google Scholar

[16] R. Lazarsfeld, Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin 2004. 10.1007/978-3-642-18808-4Search in Google Scholar

[17] A. Levin, Generalizations of Siegel’s and Picard’s theorems, Ann. of Math. (2) 170 (2009), no. 2, 609–655. 10.4007/annals.2009.170.609Search in Google Scholar

[18] A. Levin, On the Schmidt subspace theorem for algebraic points, Duke Math. J. 163 (2014), no. 15, 2841–2885. 10.1215/00127094-2827017Search in Google Scholar

[19] D. McKinnon and M. Roth, Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties, Invent. Math. 200 (2015), no. 2, 513–583. 10.1007/s00222-014-0540-1Search in Google Scholar

[20] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and new inequalities in analysis, Math. Appl. (East European Series) 61, Kluwer Academic, Dordrecht 1993. 10.1007/978-94-017-1043-5Search in Google Scholar

[21] E. I. Nochka, Defect relations for meromorphic curves, Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat. Nauk (1982), no. 1, 41–47, 79. Search in Google Scholar

[22] E. I. Nochka, On a theorem from linear algebra, Izv. Akad. Nauk Moldav. SSR Ser. Fiz.-Tekhn. Mat. Nauk (1982), no. 3, 29–33. Search in Google Scholar

[23] E. I. Nochka, On the theory of meromorphic curves, Dokl. Akad. Nauk SSSR 269 (1983), no. 3, 547–552. Search in Google Scholar

[24] E. I. Nochka, A lemma on weights, Mat. Zametki 78 (2005), no. 1, 115–124. 10.4213/mzm2566Search in Google Scholar

[25] J. Noguchi and J. Winkelmann, Nevanlinna theory in several complex variables and Diophantine approximation, Grundlehren Math. Wiss. 350, Springer, Tokyo 2014. 10.1007/978-4-431-54571-2Search in Google Scholar

[26] S. D. Quang, A generalization of the subspace theorem for higher degree polynomials in subgeneral position, Int. J. Number Theory 15 (2019), no. 4, 775–788. 10.1142/S1793042119500404Search in Google Scholar

[27] S. D. Quang, Generalizations of degeneracy second main theorem and Schmidt’s subspace theorem, Pacific J. Math. 318 (2022), no. 1, 153–188. 10.2140/pjm.2022.318.153Search in Google Scholar

[28] S. D. Quang, Some generalizations of Schmidt’s subspace theorem, preprint (2022), https://arxiv.org/abs/2212.02471. Search in Google Scholar

[29] M. Ru and P. Vojta, A birational Nevanlinna constant and its consequences, Amer. J. Math. 142 (2020), no. 3, 957–991. 10.1353/ajm.2020.0022Search in Google Scholar

[30] M. Ru and J. T.-Y. Wang, A subspace theorem for subvarieties, Algebra Number Theory 11 (2017), no. 10, 2323–2337. 10.2140/ant.2017.11.2323Search in Google Scholar

[31] M. Ru and J. T.-Y. Wang, The Ru–Vojta result for subvarieties, Int. J. Number Theory 18 (2022), no. 1, 61–74. 10.1142/S1793042122500063Search in Google Scholar

[32] M. Ru and P.-M. Wong, Integral points of P n − { 2 ⁢ n + 1 ; hyperplanes in general position } , Invent. Math. 106 (1991), no. 1, 195–216. 10.1007/BF01243910Search in Google Scholar

[33] H. P. Schlickewei, Die 𝑝-adische Verallgemeinerung des Satzes von Thue–Siegel–Roth–Schmidt, J. reine angew. Math. 288 (1976), 86–105. 10.1515/crll.1976.288.86Search in Google Scholar

[34] H. P. Schlickewei, On products of special linear forms with algebraic coefficients, Acta Arith. 31 (1976), no. 4, 389–398. 10.4064/aa-31-4-389-398Search in Google Scholar

[35] W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math. 125 (1970), 189–201. 10.1007/BF02392334Search in Google Scholar

[36] W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551. 10.2307/1970824Search in Google Scholar

[37] L. Shi, Degenerated second main theorem for holomorphic curves into algebraic varieties, Internat. J. Math. 31 (2020), no. 6, Article ID 2050042. 10.1142/S0129167X20500421Search in Google Scholar

[38] J. H. Silverman, Arithmetic distance functions and height functions in Diophantine geometry, Math. Ann. 279 (1987), no. 2, 193–216. 10.1007/BF01461718Search in Google Scholar

[39] P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Math. 1239, Springer, Berlin 1987. 10.1007/BFb0072989Search in Google Scholar

[40] P. Vojta, A refinement of Schmidt’s subspace theorem, Amer. J. Math. 111 (1989), no. 3, 489–518. 10.2307/2374670Search in Google Scholar

[41] P. Vojta, Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), no. 1, 133–181. 10.1007/s002220050092Search in Google Scholar

[42] P. Vojta, On the Nochka–Chen–Ru–Wong proof of Cartan’s conjecture, J. Number Theory 125 (2007), no. 1, 229–234. 10.1016/j.jnt.2006.10.014Search in Google Scholar

[43] P. Vojta, Birational Nevanlinna constants, beta constants, and diophantine approximation to closed subschemes, J. Théor. Nombres Bordeaux 35 (2023), no. 1, 17–61. 10.5802/jtnb.1237Search in Google Scholar

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