A generalized subspace theorem for closed subschemes in subgeneral position
aa r X i v : . [ m a t h . N T ] O c t A GENERALIZED SUBSPACE THEOREM FOR CLOSEDSUBSCHEMES IN SUBGENERAL POSITION
YAN HE AND MIN RU
Abstract.
In this paper, we extend the recent theorem of G. Heier and A.Levin [HL17] on the generalization of Schmidt’s subspace theorem and Car-tan’s Second Main Theorem in Nevanlinna theory to closed subschemes locatedin l -subgeneral position, using the generic linear combination technique due toQuang (see [Quang19]). Introduction
In recent years, there have been many efforts and developments in extending theSchmidt’s subspace theorem, as well as Cartan’s Second Main Theorem in Nevan-linna theory. The break through was made by Evertse-Ferretti [EF08] by provingthe following Theorem A (see also the result of Corvaja and Zannier [CZ04]). Weuse the standard notations in Nevanlinna theory and Diophantine approximation(see, for example, [Voj11], [Voj87], [Ru01], [Ru16] and [Ru17]). For the closedsubschemes, we use the notations in [HL17] and [RW17].
Theorem A ([EF08]) . Let X be a projective variety of dimension n defined over anumber field k . Let S be a finite set of places of k . For each v ∈ S , let D ,v , . . . , D n,v be effective Cartier divisors on X , defined over k , and in general position. Supposethat there exists an ample Cartier divisor A on X and positive integers d j,v suchthat D j,v is linearly equivalent to d j,v A (which we denote by D j,v ∼ d j,v A ) for j = 0 , . . . , n and all v ∈ S . Then, for every ǫ >
0, there exists a proper Zariski-closed subset Z ⊂ X such that for all points x ∈ X ( k ) \ Z , X v ∈ S n X j =0 d j,v λ D j,v ,v ( x ) ≤ ( n + 1 + ǫ ) h A ( x ) . Key words and phrases.
Subspace theorem, Second Main Theorem, subgeneral position.2010
Mathematics Subject Classification. Here, λ D j,v ,v is a local Weil function associated to the divisor D j,v and place v in S .The counterpart result in Nevanlinna theory was obtained by the second namedauthor [Ru09] in 2009. Theorem B ([Ru09]) . Let X be a complex projective variety and D , . . . , D q be effective Cartier divisors on X , located in general position. Suppose that thereexists an ample Cartier divisor A on X and positive integers d j such that D j ∼ d j A for j = 1 , . . . , q . Let f : C → X be a holomorphic map with Zariski dense image.Then, for every ǫ > q X j =1 d j m f ( r, D j ) ≤ exc (dim X + 1 + ǫ ) T f,A ( r )where ≤ exc means the inequality holds for all r ∈ R ≥ outside a set of finiteLebesgue measure.In order to formulate and uniformize the results for a general divisor D on X , thesecond named author introduced (see [Ru16] and [Ru17]) the notion of Nevanlinnaconstant
Nev( D ). Later, the Nevanlinna constant Nev( D ) was further developedby Ru-Vojta [RV16] to the birational Nevanlinna constant Nev bir ( L, D ), where L is a line sheaf (i.e. an invertible sheaf) over X . With the notation Nev bir ( L, D ),the following result was obtained by Ru-Vojta in [RV16].
Theorem C (Arithmetic Part, [RV16]) . Let k be a number field, and S be a finiteset of places of k . Let X be a projective variety and let D be an effective Cartierdivisor on X , both defined over k . Let L be a line sheaf on X with dim H ( X, L N ) ≥ N >
0. Then, for every ǫ >
0, there is a proper Zariski-closed subset Z of X such that the inequality m S ( x, D ) ≤ (Nev bir ( L, D ) + ǫ ) h L ( x )holds for all x ∈ X ( k ) outside of Z . Here m S ( x, D ) := P v ∈ S λ D,v ( x ) . The corresponding result in Nevanlinna theory is also obtained in [RV16]. Hereand in the following context, we only state the arithmetic results.In general Nev bir ( L, D ) is hard to compute. However, in the case when D = D + · · · + D q where D , . . . , D q are effective Cartier divisors intersecting properlyon X , Ru-Vojta [RV16] computed the Nev bir ( L, D ) in terms of the following so-called β -constant. generalized subspace theorem for closed subschemes in subgeneral position 3 Definition 1.1.
Let L be a big line sheaf and let D be a nonzero effective Cartierdivisor on a complete variety X . We define β ( L, D ) = lim inf N →∞ P m ≥ h ( L N ( − mD )) N h ( L N ) . Theorem D (Arithmetic Part, [RV16]) . Let X be a projective variety, and D , . . . , D q be effective Cartier divisors, both defined over a number field k . As-sume that D , . . . , D q intersect properly on X . Let S ⊂ M k be a finite set of placeson k . Let L be a big line sheaf on X . Then, for every ǫ >
0, there is a properZariski-closed subset Z of X such that the inequality(1) q X j =1 β ( L, D j ) m S ( x, D j ) ≤ (1 + ǫ ) h L ( x )holds for all x ∈ X ( k ) outside of Z .On the other hand, in [HL17], Heier-Levin (see also McKinnon-Roth [MR15])obtained a similar statement where β ( L, D j ) is replaced by the Seshadri constant ǫ D j ( L ) in the case when L is ample (note: Although [HL17] was listed in 2017,the statement they obtained, as noted in [HL17], indeed goes back to an earlierunpublished manuscript which they presented at 2014 meeting in Banff on Vojta’sConjectures). In addition, they obtained the results for closed subschemes as well. Definition 1.2.
Let X be a projective variety, Y be a closed subscheme of X . Let A be a nef Cartier divisor on X . The Seshadri constant ǫ Y ( A ) of Y with respect to A is a real number given by ǫ Y ( A ) = sup { γ ∈ Q : π ∗ A − γE is nef } where π : ˜ X → X is the blowing-up of X along Y with the exceptional divisor E .Note that when Y is a Cartier divisor on X , then ǫ Y ( A ) can be defined similarlywithout doing the blowing-up. Theorem E (Arithmetic Part, [HL17]) . 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 ,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 . Then, for ǫ >
0, there exists a properZariski-closed subset Z ⊂ X such that for all points x ∈ X ( k ) \ Z , X v ∈ S n X j =0 ǫ Y j,v ( A ) λ Y j,v ,v ( x ) < ( n + 1 + ǫ ) h A ( x ) . Note that, as being indicated in Ru-Vojta [RV16], when D is an effective Cartierdivisor, β ( A, D ) ≥ ǫ D ( A ) n +1 , so the above result in the divisors case is indeed a con-sequence of the result of Ru-Vojta (Note that Theorem D can also be stated inthe same form of above). However, the proof of their result is much simpler thanRu-Vojta, directly deriving from Theorem A.This paper extends the above mentioned result of Heier-Levin, i.e. Theorem E,to the case where the subschemes are in l -subgeneral position. The analytic caseis also obtained. When l = n , it is just the result of Heier-Levin. We basicallyfollow the argument of Heier-Levin [HL17], together with technique due to Quang([Quang19]) which allow us to deal with the l -subgeneral position case. Note thatthe problem of extending Theorem D to l -subgeneral position still remains open.The following is our main result. Main Theorem (Arithmetic Part) . 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 ,v , . . . , Y l,v be closed subschemes of X , defined over k , and in l -subgeneral positionwith l ≥ n . Let A be an ample Cartier divisor on X . Then, for ǫ >
0, there existsa proper Zariski-closed subset Z ⊂ X such that for all points x ∈ X ( k ) \ Z , X v ∈ S l X j =0 ǫ Y j,v ( A ) λ Y j,v ,v ( x ) < [( l − n + 1)( n + 1) + ǫ ] h A ( x ) . Combining this with Lemma 3.4 in [RW17], it gives the the following Corollary.
Corollary.
Let X be a projective variety of dimension n defined over a numberfield k . Let S be a finite set of places of k . Let Y , . . . , Y q be closed subschemes of X defined over k , in l -subgeneral position with l ≥ n . Let A be an ample Cartierdivisor on X . Then, for ǫ >
0, there exists a Zariski-closed set Z of X such thatfor all x ∈ X ( k ) \ Z , q X j =1 ǫ Y j ( A ) m S ( x, Y j ) ≤ [( l − n + 1)( n + 1) + ǫ ] h A ( x ) . Main Theorem (Analytic Part) . Let X be a complex projective variety of di-mension n . Let Y , . . . , Y q be closed subschemes of X in l -subgeneral position with l ≥ n . Let A be an ample Cartier divisor on X . Let f : C → X be a holomorphiccurve with Zariski-dense image. Then, for every ǫ > q X j =1 ǫ Y j ( A ) m f ( r, Y j ) ≤ exc [( l − n + 1)( n + 1) + ǫ ] T f,A ( r ) . generalized subspace theorem for closed subschemes in subgeneral position 5 The basic set ups of arithmetic part for closed subschemes can be found in [Sil87],the analytic analogue of which can be found in [Yam04]. In this paper, to makethe results clear, we deal with the divisor case (i.e. codim Y = 1) and the properclosed subscheme case (codim Y >
1) separately. The divisor case is a special caseof the main theorem without blowing-ups.2.
The divisor case
For a number field k , recall that M k denotes the set of places of k , and that k υ denotes the completion of k at a place υ ∈ M k . Norms k · k υ on k are normalizedso that k x k υ = | σ ( x ) | [ k υ : R ] or k p k υ = p − [ k υ : Q p ] if υ ∈ M k is an Archimedean place corresponding to an embedding σ : k ֒ → C or anon-Archimedean place lying over a rational prime p , respectively.An M k -constant is a collection ( c v ) v ∈ M k of real constants such that c v = 0 forall but finitely many v . Heights are logarithmic and relative to the number fieldused as a base field which is always denoted by k . For example, if P is a point on P nk with homogeneous coordinate [ x : · · · : x n ] in k , then h ( P ) = h O P nk (1) ( P ) = X υ ∈ M k log max {k x k υ , . . . , k x n k υ } . Let X be a non-singular projective variety over C , let D be an effective Cartierdivisor on X , and let s be a canonical section of O ( D ) (i.e., a global section s suchthat ( s = 0) = D ). Choose a Hermitian metric k ·k on O ( D ). In Nevanlinna theory,one often encounters the function(2) λ D ( x ) := − log k s ( x ) k ;this is a real-valued function on X ( C ) \ Supp D . It is linear in D (over a suitabledomain), so by linearity and continuity it can be extended to a definition of λ D fora general Cartier divisor D on X .Weil functions are counterparts to such functions in number theory. For itsdefinition and detailed properties, see [Lan83, Ch. 10] or [Voj11, Sect. 8]. Forreader’s convenience we list some properties of Weil functions for varieties anddivisors defined over k (the complex case is similar). Proposition 2.1. (see [RV16]) Let k be a number field. Let X be a normalcomplete variety, let D be a Cartier divisor on X , both defined over k . Let λ D bea Weil function for D . Then the following conditions are equivalent.(1) D is effective.(2) λ D is bounded from below by an M k -constant.(3) for all v ∈ M k , λ D,v is bounded from below.(4) there exists v ∈ M k such that λ D,v is bounded from below.
Corollary 2.2. If D − D is effective. Then λ D ≥ λ D up to an M k -constant. Proposition 2.3. (see [Voj11, Theorem 8.8]) Let X be a complete variety over anumber field k .(a) If D , D are Cartier divisors on X , let λ D , λ D be Weil functions for D , D respectively, then λ D + λ D extends uniquely to a Weil function for D + D .(b) If λ is a Weil function for a Cartier divisor D on X , and if f : X ′ → X isa morphism of k -varieties such that f ( X ′ ) Supp D , then x → λ ( f ( x )) (definedon X ′ \ Supp f ∗ D ) is a Weil function for the Cartier divisor f ∗ D on X ′ , which wedenote by λ f ∗ D .Let X be a complex projective variety and L → X be a positive line bundle.Denote by k · k a Hermitian metric in L and by ω its Chern form. Let f : C → X be a holomorphic map. We define T f,L ( r ) = Z r dtt Z | z | Let V be a normal projective variety and X ⊂ V be an irreduciblenormal subvariety of dimension n . The Cartier divisors D , . . . , D q on V are called in l -subgeneral position on X if for every choice J ⊂ { , . . . , q } with J ≤ l + 1,dim ( \ j ∈ J Supp D j ) ∩ X ≤ l − J. When V = X , then we say that the divisors D , . . . , D q on X are in l -subgeneralposition. Similar definition applies for subschemes Y , . . . , Y q .The notion “in n -subgeneral position” means the same as ”in general position”on a variety of dimension n . In the rest of the paper, by convenience, we just write D ∩ D to denote Supp D ∩ Supp D .The following is a reformulation of Theorem 1.22 in [SR13] which plays importantrole in the arguments regarding dimensions. Proposition 2.5. Let X be a projective variety, and A be an ample Cartier divisoron X . Let F ⊂ X be a proper irreducible subvariety. Then either F ⊂ A ordim( F ∩ A ) ≤ dim F − Theorem 2.6 (Arithmetic Result in the Divisors case) . Let X be a projectivevariety of dimension n defined over a number field k . Let S be a finite set of placesof k . For each v ∈ S , let D ,v , . . . , D l,v be Cartier divisors on X , defined over k ,and in l -subgeneral position with l ≥ n . 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 allpoints x ∈ X ( k ) \ Z , X v ∈ S l X j =0 ǫ D j,v ( A ) λ D j,v ,v ( x ) < [( l − n + 1)( n + 1) + ǫ ] h A ( x ) , where ǫ D j,v ( A ) are the Seshadri constants of D j,v with respect to A . We give a reformulation of a key lemma which is originally due to Quang (seeLemma 3.1 in [Quang19]). We include a proof here for the reader’s convenience. Lemma 2.7 ([Quang19], reformulated) . Let k be a number field. Let X ⊂ P Mk bean irreducible projective variety of dimension n . Let H , . . . , H l +1 be hyperplanesin P Mk which are in l -subgeneral position on X with l ≥ n . Let L , . . . , L l +1 be thenormalized linear forms defining H , . . . , H l +1 respectively. Then there exist linearforms L ′ , . . . , L ′ n +1 of M + 1 variables such that(a) L ′ = L .(b) For every t ∈ { , . . . , n + 1 } , L ′ t ∈ span k ( L , . . . , L l − n + t ) i.e., L ′ t is a k -linearcombination of L , . . . , L l − n + t .(c) Let H ′ j , j = 1 , . . . , n + 1, be the hyperplanes defined by L ′ j , j = 1 , . . . , n + 1.Then they are in general position on X . Proof. Let H ′ = H . Let W l − n +2 denote the k -vector space spanned by L , . . . , L l − n +2 . For each irreducible component Γ of H ∩ X with dim Γ = n − 1, let V Γ ⊂ W l − n +2 be the linear subspace of W l − n +2 consisting of k -linear combinationsof L , . . . , L l − n +2 that vanish entirely on Γ. Since H , . . . , H l +1 are in l -subgeneralposition on X ,dim( H ∩ H ∩ · · · ∩ H l − n +2 ∩ X ) ≤ l − ( l − n + 2) = n − < dim Γ , there must be some H j with 2 ≤ j ≤ l − n + 2 such that Γ H j , so L j is not in V Γ .Hence V Γ is a proper subspace of W l − n +2 . Observe that the choices of Γ is finite,we thus have W l − n +2 \ [ Γ V Γ = ∅ , where the union is taken over all irreducible component Γ of H ∩ X with dim Γ = n − 1. Take an L ′ ∈ W l − n +2 \ S Γ V Γ . Then L ′ does not vanish on any irreduciblecomponent Γ of H ′ ∩ X with dim Γ = n − 1. Let H ′ be the hyperplane defined by L ′ . Then, by Proposition 2.5, dim( H ′ ∩ H ′ ∩ X ) = dim( H ∩ H ′ ∩ X ) ≤ n − H ′ , H ′ are in general position on X .Now consider irreducible components Γ ′ of ( H ′ ∩ H ′ ∩ X ) with dim Γ ′ = n − 2. Let W l − n +3 be the k -vector space spanned by L , . . . , L l − n +3 and V Γ ′ ⊂ W l − n +2 be the k -linear subspace of W l − n +3 given by all k -linear combinations of L , . . . , L l − n +3 that vanishes entirely on Γ ′ . Similar to the argument above, for every such Γ ′ , V Γ ′ generalized subspace theorem for closed subschemes in subgeneral position 9 is proper linear subspace of W l − n +3 since dim( H ∩ H ∩ H , . . . , H l − n +3 ) ≤ n − < dim Γ ′ = n − 2. Thus W l − n +3 \ [ Γ ′ V Γ ′ = ∅ . Take an L ′ ∈ W l − n +3 \ S Γ ′ V Γ ′ . By the construction L ′ does not vanish on anyirreducible component Γ ′ of ( H ′ ∩ H ′ ∩ X ) with dim Γ ′ = n − 2. Let H ′ be thehyperplane defined by L ′ , then by Proposition 2.5, dim( H ′ ∩ H ′ ∩ H ′ ∩ X ) ≤ n − H ′ , H ′ , H ′ are in general position on X .Repeating the argument we obtain H ′ , . . . , H ′ n +1 which are in general positionon X . (cid:3) Proof of Theorem 2.6 . Fix ǫ > 0. Choose rational number δ > δ ( l − n + 1) + δ ( l − n + 1)( n + 1 + δ ) < ǫ. Then, for small enough positive rational number δ ′ depending on δ , δA − δ ′ D i,v is Q -ample for all i = 0 , . . . , l and v ∈ S . We fix such δ ′ . By the definition of theSeshadri constant, there exists a rational number ǫ i,v > ǫ D i,v ( A ) − δ ′ ≤ ǫ i,v ≤ ǫ D i,v ( A )and that A − ǫ i,v D i,v is Q -nef for all i = 0 , . . . , l and v ∈ S . With such choices of δ, δ ′ and ǫ i,v , we have (1 + δ ) A − ( ǫ i,v + δ ′ ) D i,v is Q -ample. Take natural number N large enough such that N (1 + δ ) A and N [(1 + δ ) A − ( ǫ i,v + δ ′ ) D i,v ] become veryample integral divisors for all i = 0 , . . . , l and v ∈ S .Fix v ∈ S , we claim that there are sections s i,v ∈ H ( X, N (1 + δ ) A − N ( ǫ i,v + δ ′ ) D i,v ) , i = 0 , . . . , l, such that their divisors div( s i,v ) , i = 0 , . . . , l , are in l -subgeneral position on X . Here we regard H ( X, N (1 + δ ) A − N ( ǫ i,v + δ ′ ) D i,v ) as a subpsace of H ( X, N (1 + δ ) A ). We prove the claim by induction. Assume that, for some j ∈ { , . . . , l } , sections s ,v , . . . , s j − ,v with the desired property have been foundand div( s ,v ) , . . . , div( s j − ,v ) , D j,v , . . . , D l,v are in l -subgeneral position (for j = 0,this reduces to the hypothesis that D ,v , . . . , D l,v are in l -subgeneral position). Inparticular, it gives(4) dim (( ∩ i ∈ I div( s i,v )) ∩ ( ∩ j ∈ J D j,v )) ≤ l − I − J, and(5) dim ( D j,v ∩ ( ∩ i ∈ I div( s i,v )) ∩ ( ∩ j ∈ J D j,v )) ≤ l − I − J − . for any subset I ⊂ { , . . . , j − } and J ⊂ { j + 1 , . . . , l } . We choose s j,v ∈ H ( X, N (1 + δ ) A − N ( ǫ j,v + δ ′ ) D j,v ) such that s j,v does not vanish entirely onany irreducible components of ( ∩ i ∈ I div( s i,v )) ∩ ( ∩ j ∈ J D j,v ) where not both I and J are empty. Then by Proposition 2.5,(6) dim (div( s j,v ) ∩ ( ∩ i ∈ I (div( s i,v )) ∩ ( ∩ j ∈ J D j,v )) ≤ l − I − J − . This means that div( s ,v ) , . . . , div( s j − ,v ) , div( s j,v ) , D j +1 ,v , . . . , D l,v are in l -subgeneral position. This finishes the induction. The claim thus is proved.Now for every v ∈ S , let F i,v := div( s i,v ) , i = 0 , . . . , l , be the divisors on X constructed above. Then, from the construction, F i,v − N ( ǫ i,v + δ ′ ) D i,v is effectivefor each i = 0 , . . . , l . Hence, by Proposition 2.1 and (3),(7) λ F i,v ,v ( x ) ≥ N ( ǫ i,v + δ ′ ) λ D i,v ,v ( x ) + O v (1) ≥ N ǫ D i,v ( A ) λ D i,v ,v ( x ) + O v (1)for x outside Supp F i,v and Supp D i,v .Denote by φ : X → P ˜ N ( k ) the canonical embedding associated to the veryample divisor N (1 + δ ) A and let H ,v , . . . , H l +1 ,v be the hyperplanes in P ˜ N ( k )with F j,v = φ ∗ H j +1 ,v for j = 0 , . . . , l . Following notation of Lemma 2.7, we de-note L ,v , . . . , L l +1 ,v the linear forms defining H ,v , . . . , H l +1 ,v respectively. ByLemma 2.7, there exists hyperplanes ˆ H ,v , . . . , ˆ H n +1 ,v with defining linear formsˆ L t,v , t = 1 , . . . , n + 1, such that ˆ L ,v = L ,v , ˆ L t,v ∈ span k ( L ,v , . . . , L l − n + t,v ) for t = 2 , . . . , n + 1 and φ ∗ ˆ H ,v , . . . , φ ∗ ˆ H n +1 ,v are located in general position on X .Applying Theorem A to φ ∗ ˆ H ,v , . . . , φ ∗ ˆ H n +1 ,v , we conclude that there exists aZariski-closed set Z such that for all x ∈ X ( k ) \ Z ,(8) X v ∈ S n +1 X i =1 λ φ ∗ ˆ H i,v ,v ( x ) ≤ [( n + 1) + δ ] h N (1+ δ ) A ( x ) . On the other hand, fixing P = φ ( x ) ∈ P ˜ N ( k ), we reorder H ,v , . . . , H l +1 ,v such that k L ,v ( P ) k v ≤ k L ,v ( P ) k v ≤ · · · ≤ k L l +1 ,v ( P ) k v . For t = 2 , . . . , n + 1, using the factthat ˆ L t,v ∈ span k ( L ,v , . . . , L l − n + t,v ), we have k ˆ L t,v ( P ) k v ≤ C v max ≤ j ≤ l − n + t k L j,v ( P ) k v = k L l − n − t,v ( P ) k v for some constant C v > 0. Thus, by the definition of λ ˆ H t ,v ( P ), λ ˆ H t,v ,v ( P ) ≥ λ H l − n − t,v ,v ( P ) + O v (1) . generalized subspace theorem for closed subschemes in subgeneral position 11 Therefore l +1 X j =1 λ H j,v ,v ( P ) = l − n +1 X j =1 λ H j,v ,v ( P ) + l +1 X j = l − n +2 λ H j,v ,v ( P ) ≤ l − n +1 X j =1 λ H j,v ,v ( P ) + n +1 X t =2 λ ˆ H t,v ,v ( P ) + O v (1) ≤ ( l − n + 1) λ H ,v ,v ( P ) + n +1 X t =2 λ ˆ H t,v ,v ( P ) + O v (1) ≤ ( l − n + 1) n +1 X i =1 λ ˆ H i,v ,v ( P ) + O v (1) . (9)Noticing that P = φ ( x ) and using the fact that λ H j,v ,v ( φ ( x )) = λ φ ∗ H j,v ,v ( x ) = λ F j − ,v ,v ( x ), it gives, by combining (8) and (9), we obtain X v ∈ S l X i =0 λ F i,v ,v ( x ) = X v ∈ S l +1 X j =1 λ H j,v ,v ( P ) ≤ ( l − n + 1) X v ∈ S n +1 X i =1 λ ˆ H i,v ,v ( P ) + O (1) ≤ ( l − n + 1)[( n + 1) + δ ] h N (1+ δ ) A ( x ) . (10)This, together with (7), gives N X v ∈ S l X j =0 ǫ D j,v ( A ) λ D j,v ,v ( x ) ≤ X v ∈ S l X i =0 λ F i,v ,v ( x ) ≤ ( l − n + 1)[( n + 1) + δ ] h N (1+ δ ) A ( x ) + O (1)for x ∈ X ( k ) \ Z . Note that h N (1+ δ ) A ( x ) = N (1 + δ ) h A ( x ), hence we can rewritethe above inequality as X v ∈ S l X j =0 ǫ D j,v ( A ) λ D j,v ,v ( x ) ≤ ( l − n + 1)( n + 1 + δ )(1 + δ ) h A ( x ) . Recall our choice of δ , we get for all x ∈ X ( k ) \ Z , X v ∈ S l X j =0 ǫ D j,v ( A ) λ D j,v ,v ( x ) ≤ [( l − n + 1)( n + 1) + ǫ ] h A ( x ) . The theorem is thus proved.Note that Lemma 2.7 also holds for the field C case. So the above argumenttogether, with Theorem B to replace Theorem A (note that the statement of The-orem B is slightly different from Theorem A, but we can obtain a statement whichexactly corresponds to Theorem B, see Theorem 1.8 in [HL17]), which derives thefollowing analytic result. Theorem 2.8 (Analytic Part) . Let X be a complex projective variety of dimension n . Let D , . . . , D q be effective Cartier divisors located in l -subgeneral position on X with l ≥ n . Let A be an ample Cartier divisor on X . Let f : C → X be aholomorphic map with Zariski dense image. Then, for ǫ > q X j =1 ǫ D j ( A ) m f ( r, D j ) ≤ exc [( l − n + 1)( n + 1) + ǫ ] T f,A ( r ) . The subscheme case We follow the notation in [RW17] or [HL17]. Let Y be a closed subscheme on aprojective variety V defined over k . Then one can associate to each place v ∈ M k a function λ Y,v : V \ Supp( Y ) → R satisfying some functorial properties (up to a constant) described in [Sil87, Theorem2.1]. Intuitively, for each P ∈ V, v ∈ M k λ Y,v ( P ) = − log( v -adic distance from P to Y ) . The starting point to study such arithmetic distance functions on subschemes is tounderstand the natural operations (i.e. addition, intersection, image and inverseimage under morphism) on subschemes, which are parallel with the case of divisors.The key observation here is that we can identify a closed subscheme Y of V withits ideal sheaf I Y , and such operations are naturally defined in terms of operationson ideal sheaves. We briefly summarize them as follows, details can be found inSection 2, [Sil87]. Let Y, Z be closed subschemes of V .(i) The sum of Y and Z , denoted by Y + Z is the subscheme of V with idealsheaf I Y I Z .(ii) The intersection of Y and Z , denoted by Y ∩ Z , is the subscheme of V withideal sheaf I Y + I Z .(iii) The union of Y and Z , denoted by Y ∪ Z , is the subscheme of V with idealsheaf I Y ∩ I Z .(iv) Let φ : W → V be a morphism of varieties. The inverse image of Y is thesubscheme of W with ideal sheaf φ − I Y · O W , denoted by φ ∗ Y .The following lemma indicates the existence of local Weil functions for closedsubschemes: generalized subspace theorem for closed subschemes in subgeneral position 13 Lemma 3.1 (Lemma 2.2 in [Sil87]) . Let Y be a closed subscheme of V . Thereexist effective Cartier divisors D , · · · , D r such that Y = ∩ ri =1 D i . Definition 3.2. Let k be a number field, and M k be the set of places on k . Let V be a projective variety over k and let Y ⊂ V be a closed subscheme of V . Wedefine the (local) Weil function for Y with respect to v ∈ M k as λ Y,v = min ≤ i ≤ r { λ D i ,v } , (11)when Y = ∩ ri =1 D i (such D i exist according to the above lemma), where λ D i ,v is theWeil function for Cartier divisors appeared in previous section. Hence in particularif the closed subscheme is a Cartier divisor then the definition coincides with theone given in discussion of divisor case.The behavior of Weil functions for closed subschemes have some similarity withthe case of divisors in the sense as follows: Theorem 3.3 (Theorem 2.1 in [Sil87]) . Let Y, Z be closed subschemes of V . Thenup to O v (1):(i) λ Y ∩ Z,v = min { λ Y,v , λ Z,v } . (ii) λ Y + Z,v = λ Y,v + λ Z,v . (iii) If Y ⊂ Z then λ Y,v ≤ λ Z,v .Furthermore, let φ : W → V be morphism of varieties, one has(iv) λ φ ∗ Y,v ( P ) = λ Y,v ( φ ( P )) for P ∈ W \ Supp φ ∗ Y . Definition 3.4. Let X be a projective variety. Y be a closed subscheme of X corresponding to a coherent sheaf of ideals I . S be the sheaf of graded algebras L d ≥ I d with convention I = O X and I d be the d -th power of I . Then thescheme Proj S is called the blowing-up of X with respect to I or blowing-up of X along Y .The functoriality of Weil function stated in (iv) of above theorem, is of particularimportance when we deal with blowing-ups, hence we reformulate it as a lemma. Lemma 3.5 (Lemma 2.5.2 in [Voj87]; Theorem 2.1(h) in [Sil87]) . Let Y be a closedsubscheme of V , and let ˜ V be the blowing-up of V along Y with exceptional divisor E . Then λ Y,v ( π ( P )) = λ E,v ( P ) + O v (1) for P ∈ ˜ V \ Supp E . The strategy to deal with closed subschemes is to look at blowing-ups and relateto the case of divisors using functoriality of Weil functions and Height functions.However, to apply the method in divisor case we need to obtain linear equivalentample Cartier divisors in l -subgeneral position, for which it’s convenient to go backdownstairs to avoid dealing with the influence of exceptional divisors. The follow-ing lemma allows us to find Cartier divisors downstairs with desired intersectionproperty. Lemma 3.6 (See Lemma 5.4.24 in [Laz17]) . Let X be projective variety, I bea coherent ideal sheaf. Let π : ˜ X → X be the blowing-up of I with exceptionaldivisor E . Then there exists an integer p = p ( I ) with the property that if p ≥ p ,then π ∗ O ˜ X ( − pE ) = I p , and moreover, for any divisor D on X , H i ( X, I p ( D )) = H i ( ˜ X, O ˜ X ( π ∗ D − pE ))for all i ≥ Proof of The Main Theorem . Denote by I i,v the ideal sheaf of Y i,v , π i,v : ˜ X i,v → X the blowing-up of X along Y i,v , and E i,v the exceptional divisor on ˜ X i,v . Fix realnumber ǫ > 0. Choose rational number δ > δ ( l − n + 1) + δ ( l − n + 1)( n + 1 + δ ) < ǫ. Then for small enough positive rational number δ ′ depending on δ , δπ ∗ A − δ ′ E i,v is Q -ample on ˜ X i,v for all i ∈ { , . . . , l } and v ∈ S . By the definition of Seshadriconstant, there exists a rational number ǫ i,v > ǫ i,v + δ ′ ≥ ǫ Y i,v ( A )and π ∗ i,v A − ǫ i,v E i,v is Q -nef on ˜ X i,v for all 0 ≤ i ≤ l, v ∈ S . With such choices, wehave (1 + δ ) π ∗ i,v A − ( ǫ i,v + δ ′ ) E i,v is ample Q -divisor on ˜ X i,v for all i, v . Let N ≫ N (1 + δ ) π ∗ i,v A and N [(1 + δ ) π ∗ i,v A − ( ǫ i,v + δ ′ ) E i,v ]are very ample integral divisors on ˜ X i,v for all 0 ≤ i ≤ l and v ∈ S .Fix v ∈ S , like the special divisor case in the above section, we claim that wecan construct Cartier divisors F ,v , . . . , F l,v on X which are located in l -subgeneralposition, such that F i,v ∼ N (1 + δ ) A and π ∗ i,v F i,v ≥ N ( ǫ i,v + δ ′ ) E i,v on ˜ X i,v forall 0 ≤ i ≤ l . This can be done inductively. Assume F ,v , . . . , F j − ,v have been generalized subspace theorem for closed subschemes in subgeneral position 15 constructed so that F i,v ∼ N (1 + δ ) A and π ∗ i,v F i,v ≥ N ( ǫ i,v + δ ′ ) E i,v on ˜ X i,v for all0 ≤ i ≤ j − 1, and that F ,v , . . . , F j − ,v , Y j,v , . . . , Y l,v are in l -subgeneral position on X (for j = 0, this reduces to the hypothesis that Y ,v , . . . , Y l,v are in l -subgeneralposition). To find F j,v , we let ˜ F ( j ) i,v = π ∗ j,v F i,v , i = 0 , . . . , j − 1, and ˜ Y ( j ) i,v = π ∗ j,v Y i,v for i = j + 1 , . . . , l . Since, in particular, F ,v , . . . , F j − ,v , Y j +1 ,v , . . . , Y l,v are in l -subgeneral position on X , and by noticing that π − j,v is isomorphism outside of Y j,v ,we know that ˜ F ( j )0 ,v , . . . , ˜ F ( j ) j − ,v , ˜ Y ( j ) j +1 ,v , . . . , ˜ Y ( j ) l,v are in l -subgeneral position on ˜ X j,v outside of E j,v . It is thus reduced to the construction in the divisors case, and bythe argument in the divisors case, there are sections˜ s j,v ∈ H ( ˜ X j,v , O ˜ X j,v ( N ((1 + δ ) π ∗ j,v A − ( ǫ j,v + δ ′ ) E j,v ))) , such that ˜ F ( j )0 ,v , . . . , ˜ F ( j ) j − ,v , div(˜ s j,v ) , ˜ Y ( j ) j +1 ,v , . . . , ˜ Y ( j ) l,v are in l -subgeneral position on˜ X j,v outside of E j,v , where we regard H ( ˜ X j,v , O ˜ X j,v ( N ((1 + δ ) π ∗ j,v A − ( ǫ j,v + δ ′ ) E j,v ))) as a subspace of H ( ˜ X j,v , O ˜ X j,v ( N ((1 + δ ) π ∗ j,v A ))) . On the other hand,by Lemma 3.6, we have, for N big enough, H ( X, O X ( N (1+ δ ) A ) ⊗I N ( ǫ j,v + δ ′ ) j,v ) = H ( ˜ X j,v , O ˜ X j,v ( N ((1+ δ ) π ∗ j,v A − ( ǫ j,v + δ ′ ) E j,v ))) . Therefore there is an effective divisor F j,v ∼ N (1 + δ ) A on X such that div(˜ s j,v ) = π ∗ j,v F j,v . Since ˜ s j,v ∈ H ( ˜ X j,v , O ˜ X j,v ( N ((1 + δ ) π ∗ j,v A − ( ǫ j,v + δ ′ ) E j,v ))) , we have π ∗ j,v F j,v ≥ N ( ǫ j,v + δ ′ ) E j,v on ˜ X j,v . To complete the induction, it remains toshow that F ,v , . . . , F j − ,v , F j,v , Y j +1 ,v , . . . , Y l,v are in l -subgeneral position on X .Since ˜ F ( j )0 ,v , . . . , ˜ F ( j ) j − ,v , div(˜ s j,v ) , ˜ Y ( j ) j +1 ,v , . . . , ˜ Y ( j ) l,v are in l -subgeneral position on ˜ X j,v outside of E j,v , and π j,v is an isomorphism above the complement of Y j,v , it is clearthat F ,v , . . . , F j − ,v , F j,v , Y j +1 ,v , . . . , Y l,v are in l -subgeneral position on X outside Y j,v . The full statement now follows from combining this with the fact that Y j,v is in l -subgeneral position with F ,v , . . . , F j − ,v , Y j +1 ,v , . . . , Y l,v . This meets therequirement. So the claim holds by induction.Following the claim, for every v ∈ S , we get linearly equivalent Cartier divisors F i,v ∼ N (1 + δ ) A , i = 0 , . . . , l in l -subgeneral position on X . By the same way inderiving (10), we get(13) X v ∈ S l X i =0 λ F i,v ,v ( x ) ≤ ( l − n + 1)[( n + 1) + δ ] h N (1+ δ ) A ( x )on X ( k ) \ Z where Z is a proper Zariski-closed subset of X . To relate with Y j,v ,by the fact that π ∗ j,v F j,v ≥ N ( ǫ j,v + δ ′ ) E j,v on ˜ X i,v , by applying Proposition 2.1, Lemma 3.5, and (12), for all P ∈ ˜ X j,v \ Supp E j,v , N ǫ Y j,v ( A ) λ Y j,v ,v ( π j,v ( P )) ≤ N ( ǫ j,v + δ ′ ) λ Y j,v ,v ( π j,v ( P ))= N ( ǫ j,v + δ ′ ) λ E j,v ,v ( P ) ≤ λ π ∗ j,v F j,v ,v ( P ) = λ F j,v ,v ( π j,v ( P )) . This, together with (13), gives N X v ∈ S l X j =0 ǫ Y j,v ( A ) λ Y j,v ,v ( x ) ≤ X v ∈ S l X j =0 λ F j,v ( x ) ≤ ( l − n + 1)[( n + 1) + δ ] h N (1+ δ ) A ( x )= ( l − n + 1)[( n + 1) + δ ] N (1 + δ ) h A ( x )(14)on X ( k ) \ Z . By the choice of δ , the result follows.In the complex case, we use the notations in [RW17]. In particular, for a complexprojective variety X and a holomorphic map f : C → X with f ( C ) Y , we definethe proximity function m f ( r, Y ) = Z π λ Y ( f ( re iθ )) dθ π . The above argument also works, by replacing Theorem A with Theorem B, we canprove the analytic result of the Main Theorem stated in the introduction section. References [CZ04] Pietro Corvaja and Umberto Zannier. On a general thue’s equation. American Journalof Mathematics , 126(5):1033–1055, 2004.[EF08] Jan-Hendrik Evertse and Roberto G Ferretti. A generalization of the subspace theo-rem with polynomials of higher degree. In Diophantine approximation , pages 175–198.Springer, 2008.[HL17] Gordon Heier and Aaron Levin. A generalized schmidt subspace theorem for closed sub-schemes. arXiv preprint arXiv:1712.02456 , 2017.[Lan83] Serge Lang. Fundamentals of diophantine geometry . Springer-Verlag New York, 1983.[Laz17] Robert K Lazarsfeld. 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Algebro-geometric version of Nevanlinna’s lemma on logarithmicderivative and applications. In Nagoya Math. J. , 173 (2004), 23–63. Department of MathematicsUniversity of HoustonHouston, TX 77204, U.S.A. E-mail address : [email protected] Department of MathematicsUniversity of HoustonHouston, TX 77204, U.S.A. E-mail address ::