DSpace at VNU: THE SECOND MAIN THEOREM FOR HYPERSURFACES

Kyushu J Math 65 (2011), 219–236 doi:10.2206/kyushujm.65.219 THE SECOND MAIN THEOREM FOR HYPERSURFACES DO DUC THAI and NINH VAN THU (Received November 2009) Abstract The purpose of this article is twofold The first is to show the Second Main Theorem for degenerate holomorphic curves into Pn (C) with hypersurface targets located in n-subgeneral position The second is to show the Second Main Theorem with truncated counting functions for nondegenerate holomorphic curves into Pn (C) with hypersurface targets in general position Finally, by applying the above result, a unicity theorem for algebraically nondegenerate curves into P2 (C) with hypersurface targets in general position is also given Introduction and main results Let f : C → Pn (C) be a holomorphic map Let f˜ = (f0 , , fn ) be a reduced representation of f , where f0 , , fn are entire functions on C and have no common zeros The Nevanlinna–Cartan characteristic function Tf (r) is defined by Tf (r) = where 2π 2π log f˜(reiθ ) dθ, f˜(z) = max{|f0 (z)|, , |fn (z)|} The above definition is independent, up to an additive constant, of the choice of a reduced representation of f Let D be a hypersurface in Pn (C) of degree d Let Q be the homogeneous polynomial (form) of degree d defining D The proximity function mf (r, D) is defined as 2π mf (r, D) = log f˜(reiθ ) d Q dθ , |Q(f˜)(reiθ )| 2π where Q is the sum of the absolute values of the coefficients of Q The above definition is independent, up to an additive constant, of the choice of a reduced representation of f To define the counting function, let nf (r, D) be the number of zeros of Q(f˜) in the disk |z| < r, counting multiplicity The counting function is then defined by 2π Nf (r, D) = nf (t, D) − nf (0, D) dt + nf (0, D) log r t 2000 Mathematics Subject Classification: Primary 32H30; Secondary 32H04, 32H25, 14J70 Keywords: holomorphic curves; algebraic degeneracy; defect relation; Nochka weight c 2011 Faculty of Mathematics, Kyushu University 220 Do Duc Thai and Ninh Van Thu Note that 2π dθ log |Q(f˜)(reiθ )| + O(1) 2π The Poisson–Jensen formula implies the following Nf (r, D) = F IRST M AIN T HEOREM Let f : C → Pn (C) be a holomorphic map, and let D be a hypersurface in Pn (C) of degree d If f (C) ⊂ D, then for every real number r with < r < +∞, mf (r, D) + Nf (r, D) = dT f (r) + O(1), where O(1) is a constant independent of r Recall that hypersurfaces D1 , , Dq in Pn (C)(q > n) are said to be in general position Supp(Djk ) = ∅ for any distinct j1 , , jn+1 ∈ {1, , q} if In [8], Ru showed the Second Main Theorem (SMT, for short) for algebraically nondegenerate holomorphic curves into Pn (C) and fixed hypersurface targets in general position in Pn (C) As a corollary of this theorem, he proved the Shiffman conjecture for algebraically nondegenerate holomorphic curves into Pn (C) and fixed hypersurfaces in general position in Pn (C) Later, Dethloff and Tan [4] showed a SMT for algebraically nondegenerate meromorphic maps of Cm into Pn (C) and q slowly moving hypersurfaces targets in Pn (C) (q ≥ n + 2) in (weakly) general position (cf the detailed definitions in [4]) The following question arose naturally at this moment: is there the SMT for degenerate holomorphic curves into Pn (C) and hypersurface targets in Pn (C), i.e., a theorem of Cartan–Nochka type? The first aim of this article is to show the SMT for degenerate holomorphic curves into Pn (C) with hypersurface targets located in n-subgeneral position (cf Definition 2.5) Namely, we show the following ∩n+1 k=1 T HEOREM 1.1 (SMT for algebraically degenerate holomorphic curves) Let f : C → Pn (C) be a holomorphic map whose image is contained in some k-dimensional subspace but not in q any subvariety of dimension lower than k Let {Dj }j =1 be hypersurfaces in Pn (C) of degree dj , located in n-subgeneral position in that subspace Then for every > 0, q j =1 dj−1 mf (r, Dj ) ≤ (2n − k + + )Tf (r) As usual, by notation ‘ P ’ we mean that the assertion P holds for all r ∈ [0, ∞) excluding a Borel subset E of the interval [0, ∞) with E dr < ∞ We remark that in the above-mentioned original SMT of Ru [8], there is no truncated counting function that can be used in the study of uniqueness problems Later, An and Phuong [1] and Dethloff and Tan [4] gave truncated counting functions in this theorem Namely, they showed the following q Let f : C → Pn (C) be an algebraically nondegenerate holomorphic map, and let {Dj }j =1 be fixed hypersurfaces in Pn (C) of degree dj in general position Let d be the least common multiple of the dj Then, for every < < 1, q (q − n − − )Tf (r) ≤ j =1 where: (Mj ) N (r, div(f, Dj )), dj The Second Main Theorem 221 M1 = · · · = Mq ≥ 2d[2n (n + 1)n(d + 1) −1 ]n (see An and Phuong [1]); − dj dj · n+N n + 1, where (ii) Mj ≤ d n N = d · [2(n + 1)(2 − 1)(nd + 1) −1 + n + 1] (see Dethloff and Tan [4]) Here and in what follows [x] = min{n ∈ Z : n ≥ x} (the ceiling function) and N (α) (r, ν) is the counting function with the truncation level to α of divisor ν (i) However, their truncation level is very big The second aim of this article is to give a better truncation in their SMT To state our second result, we need some notation Let N be any positive integer For each i = (i1 , , in ) ∈ Nn with σ (i) = i1 + · · · + in ≤ N/d, denote by (i) the number of n-tuples (j1 , , jn ) ∈ Nn such that j1 + · · · + jn ≤ N − dσ (i) and ≤ j1 , , jn ≤ d − Let = (1/n) σ (i)≤N/d σ (i) (i) and M = n+N Here is our n result T HEOREM 1.2 Let f : C → Pn (C) be an algebraically nondegenerate holomorphic curve, q and let {Dj }j =1 be hypersurfaces in Pn (C) of degree dj in general position Let d be the least common multiple of the dj Then dq − MN Tf (r) ≤ N ([M 2/ ]) (r, div(f, D)) + O(ln Tf (r)), q j =1 (d/dj ) div(f, Dj ) where D = D1 ∪ · · · ∪ Dq and div(f, D) = In the above result, we have the following explicit estimate C OROLLARY 1.3 Let f : C → Pn (C) be an algebraically nondegenerate holomorphic q curve, and let {Dj }j =1 be hypersurfaces in Pn (C) of degree dj in general position Let d be the least common multiple of the dj Then, for every < ≤ (n + 1)/d, d(q − n − − )Tf (r) ≤ N (L) (r, div(f, D)) + O(ln Tf (r)), where L = 2den d(n + 1)2 −1 and n−1 q div(f, D) = j =1 , D = D1 ∪ · · · ∪ Dq d div(f, Dj ) dj In the last part of this article, by applying the SMT with explicit truncated counting functions for nondegenerate holomorphic curves into P2 (C) and hypersurface targets located in general position, a unicity theorem for algebraically nondegenerate curves into P2 (C) with hypersurface targets in general position is also given T HEOREM 1.4 Let f, g : C → P2 (C) be algebraically nondegenerate curves, and let q Qj j =1 be homogeneous polynomials of degree dj in general position in P2 (C) Let d be the least common multiple of the dj Assume that: (i) f = g on q −1 (Q ) ∪ g −1 (Q )); j j j =1 (f (ii) f −1 (Qi ) ∩ f −1 (Qj ) = ∅ for i = j 222 Do Duc Thai and Ninh Van Thu Then f ≡ g for each q≥ 4(d + 1)(2d + 1) 4(d + 1)2 (2d + 1)2 + + d d(d + 3d + 4) d + 3d + Second Main Theorem for the degenerate case First of all, we need the following general form of the SMT for holomorphic curves intersecting hyperplanes They are stated and proved in [6] and [7] T HEOREM 2.1 (See [6, Theorem 2.1]) Let F = [f0 : · · · : fn ] : C → Pn (C) be a holomorphic map whose image is not contained in any proper linear subspace Let H1 , , Hq be arbitrary hyperplanes in Pn (C) Let Lj , ≤ j ≤ q, be the linear forms defining H1 , , Hq Then, for every > 0, F˜ (reiθ ) Lβ(t ) − (n + 1) log F˜ (reiθ ) |Lβ(t ) (F˜ (reiθ ))| n 2π max log β∈K t =0 dθ < T (r, F ), 2π where the maximum is taken over all subsets K of {1, , q} such that the linear forms Lj , j ∈ K, are linearly independent, and Lj is the maximum of the absolute values of the coefficients in Lj T HEOREM 2.2 (See [7, Theorem A3.1.3]) Let f = [f0 : · · · : fn ] : C → Pn (C) be a holomorphic map whose image is not contained in any proper linear subspace Let H1 , , Hq be arbitrary hyperplanes in Pn (C) Let Lj , ≤ j ≤ q, be the linear forms defining H1 , , Hq Denote by W (f0 , , fn ) the Wronskian of f0 , , fn Then for every > 0, n 2π max log β∈K t =0 f˜(reiθ ) Lβ(t ) dθ + NW (r, 0) |Lβ(t )(F˜ (reiθ ))| 2π < (n + 1)Tf (r) + n(n + 1) (ln Tf (r) + (1 + ) ln+ (ln Tf (r))) + O(1), (1) where the maximum is taken over all subsets K of {1, , q} such that the linear forms Lj , j ∈ K, are linearly independent, and Lj is the maximum of the absolute values of the coefficients in Lj Hyperplanes b1 , , bq ∈ Pk (C)∗ are said to be in n-subgeneral position if, for every ≤ j1 < · · · < jn+1 ≤ q, the linear span of bj1 , , bjn+1 is Ck+1 Recall the following lemma about Nochka weights; for details, see [10] L EMMA 2.3 (Nochka) Let B = {bj ∈ Pk (C)∗ , ≤ j ≤ q} be the set of hyperplanes in n-subgeneral position with 2n − k + ≤ q For each ∅ = P ⊂ {1, , q}, let L(P ) be the linear space generated by {b˜j | j ∈ P }, where b˜ j is a reduced representation of bj Then there exists numbers ω1 , , ωq ∈ (0, 1], called the Nochka weights, and a real number ≥ 1, called the Nochka constant, satisfying the properties: (i) if j ∈ {1, 2, , q}, then < ωj ≤ 1; q (ii) q − 2n + k − ≤ ( j =1 ωj − k − 1); (iii) if ∅ = P ⊂ {1, , q} with #P ≤ n + 1, then j ∈P ωj ≤ dim L(P ); (iv) ≤ (n + 1)/(k + 1) ≤ ≤ (2n − k + 1)/(k + 1); The Second Main Theorem (v) 223 given E1 , , Eq , real numbers ≥ 1, and given any Y ⊂ {1, , q} with < #Y ≤ n + 1, there exists a subset M of Y such that {b˜j }j ∈M is a basis for L(Y ) and ω j ∈Y Ej j ≤ Ej j ∈M L EMMA 2.4 Let Q1 , , Qn be n homogeneous polynomials of the same degree d > in C[x1 , , xn ] If the Jacobian ∂(Q1 , , Qn )/∂(x1 , , xn ) ≡ on Cn , then Q1 , , Qn have a common zero (x1 , , xn ) = (0, , 0) Proof Let S be a variety defined by S = {x = (x1 , , xn ) ∈ Cn | Q3 (x) = · · · = Qn (x) = 0} If dim S > 2, then dim{x = (x1 , , xn ) ∈ Cn | Qj (x) = 0, ≤ j ≤ n} > Therefore, Q1 , , Qn have a common zero (x1 , , xn ) = (0, , 0) So, we may assume that dim S = Denote by S˜ the set of all smooth points in S Let us consider an arbitrary smooth ˜ Without loss of generality, we may assume that there exist a neighborhood point x0 ∈ S U of x0 in S and a homeomorphism h : C2 → U ∩ S such that (U, h) is a chart on S and ∂(Q3 , , Qn )/∂(x3 , , xn ) = on U ∩ S It means that Qj (h(u, v)) = for every (u, v) ∈ C2 and for every ≤ j ≤ n Let Q˜ (u, v) := Q1 (h(u, v)) and Q˜ (u, v) := Q2 (h(u, v)) for every (u, v) ∈ C2 Note that n ˜l ∂Q ∂Ql ∂hk = , ∂v ∂xk ∂v k=1 n ˜l ∂Ql ∂hk ∂Q = , ∂u ∂xk ∂u k=1 n k=1 n ∂Qj ∂hk = 0, ∂xk ∂u k=1 ∂Qj ∂hk = 0, ∂xk ∂v l = 1, 2, j = 3, , n By using Euler’s formula and ∂(Q1 , , Qn )/∂(x1 , , xn ) ≡ on Cn , it follows that Q˜ (u, v) Q˜ (u, v) ··· ∂ Q˜ ∂u (u, v) ∂ Q˜ ∂u (u, v) ··· ∂Q1 (h(u,v)) ∂x3 ∂Q2 (h(u,v)) ∂x3 ∂Q3 (h(u,v)) ∂x3 ··· ∂Qn (h(u,v)) ∂x3 ∂Q1 (h(u,v)) ∂xn ∂Q2 (h(u,v)) ∂xn ∂Q3 (h(u,v)) ∂xn ··· ∂Qn (h(u,v)) ∂xn Since ∂(Q3 , , Qn )/∂(x3 , , xn ) = on U ∩ S, Q˜ Q˜ ∂ Q˜ ∂u ∂Q2 ∂u Q˜ Q˜ ∂ Q˜ ∂v ∂Q2 ∂v ≡ on C2 Similarly, we also have ≡ on C2 ≡0 on C2 (2) 224 Do Duc Thai and Ninh Van Thu Therefore, there exists a constant C such that Q˜ (u, v) = C Q˜ (u, v) for every (u, v) ∈ C2 ; that is, Q1 (x) = CQ2 (x) for all x ∈ U ∩ S Since x0 is an arbitrary smooth point in S, ˜ Hence, Q1 = CQ2 on S since S˜ is dense in S Consequently, Q1 (x) = CQ2 (x) for all x ∈ S there is a common zero (x1 , , xn ) = (0, , 0) of Q1 , , Qn This completes the proof ✷ Let D1 , , Dq , q > n be hypersurfaces in Pk (C) Denote by Qj the homogeneous polynomial (form) of degree dj (smallest) defining Dj , ≤ j ≤ q For each ≤ j ≤ q and p for each p ∈ Pk (C), the hyperplane Hj is defined by k l=0 ∂Qj (p)zl = ∂zl Let T be the set of all injective maps α : {0, , n} → {1, , q} For each α ∈ T , denote by Aα[k] (p) the matrix ∂Qα(i) (p) 0≤i≤n ∂zj 0≤j ≤k Denote by Mα the set of all p ∈ Pk (C) such that rank Aα[k] (p) < k + and let M = ∪α∈T Mα Let K be the set of all injective maps β : {0, , k} → {1, , q} such that ∩kl=0 Supp(Dβ(l) ) = ∅ For each β ∈ K, let M˜ β := p ∈ Pk (C) ∂(Qβ(0) , , Qβ(k) ) (p) = ∂(z0 , , zk ) Let M˜ = Pk (C) if K = ∅ and let M˜ = ∩β∈K M˜ β if otherwise Definition 2.5 Hypersurfaces D1 , , Dq , q > n, in Pk (C) are said to be in n-subgeneral position if M˜ \ M = ∅ Definition 2.6 Let D1 , , Dq , q > n, be hypersurfaces in Pn (C) Let V be a linear subspace of Pn (C) {D1 , , Dq } are called in n-subgeneral position in V if Dˆ i1 , , Dˆ it are in n-subgeneral position, where Dˆ is := Dis ∩ V if Dis ∩ V = V Example 2.7 Let D1 , , Dq , q > n, be hypersurfaces of the same degree d ≥ in Pk (C) Assume that: (i) Dj1 ∩ · · · ∩ Djn+1 = ∅ for any distinct j1 , , jn+1 ∈ {1, 2, , q}, (ii) for any distinct i0 , , ik ∈ {1, , q} such that Qi0 , , Qik are linearly independent, Di0 ∩ · · · ∩ Dik = ∅ Then D1 , , Dq are in n-subgeneral position in Pk (C) Proof By Lemma 2.4, M is a proper analytic set in Pk (C) On the other hand, M˜ = Pk (C) Therefore, M˜ \ M = ∅ ✷ By Definition 2.5, we have the following L EMMA 2.8 Assume that D1 , , Dq , q > n, are hypersurfaces in Pk (C), located in nsubgeneral position Then for each p ∈ M˜ \ M and for any distinct i0 , , ik ∈ {1, , q}, p p Di0 ∩ · · · ∩ Dik = ∅ iff Hi0 , , Hik are linearly independent The Second Main Theorem 225 Proof of Theorem 1.1 Without loss of generality, we may assume that f : C → Pk (C) ⊂ Pn (C) is a holomorphic map So f : C → Pk (C) is a nondegenerate holomorphic map We also assume that q ≥ 2n − k + Let f˜ = (f0 , , fk ) be a reduced representation of f , where f0 , , fk are entire functions on C and have no common zeros For simplicity, we just write f˜ as f Let D1 , , Dq be hypersurfaces in Pn (C), located in general position Let Qj , ≤ j ≤ q, be the homogeneous polynomials in C[x0 , , xn ] of degree dj defining d/dj Dj Replacing Qj by Qj if necessary, where d is the least common multiple of the dj , we may assume that Q1 , , Qq have the same degree of d Let Dˆ j = Dj ∩ Pk (C) and let ˆ Q(z) = Q(z) for z ∈ Pk (C) Without loss of generality, we may assume that Dˆ , , Dˆ q are hypersurfaces in Pk (C) Then Dˆ , , Dˆ q are in n-subgeneral position in Pk (C) By Lemma 2.8, there exist hyperplanes H1 , , Hq , located in n-subgeneral position in Pk (C) such that, for any distinct i0 , , ik ∈ {1, , q}, if Di0 ∩ · · · ∩ Dik = ∅ then Hi0 , , Hik are linearly independent Let ω1 , , ωq be the Nochka weights and let be the Nochka constant associated to the hyperplanes H1 , , Hq by Lemma 2.3 Given z ∈ C, there exists a renumbering {i1 , , iq } of the indices {1, , q} such that ˆ i2 ◦ f (z)| ≤ · · · ≤ |Qˆ iq ◦ f (z)| |Qˆ i1 ◦ f (z)| ≤ |Q (3) ˆ , , Qˆ q are in n-subgeneral position and by Hilbert’s Nullstellensatz, for any Since Q integer l (0 ≤ l ≤ n), there is an integer ml ≥ d such that n+1 xlml = bjl (x0 , , xk )Qˆ ij (x0 , , xk ), j =1 where bjl , ≤ j ≤ n + 1, ≤ l ≤ k, are the homogeneous forms with coefficients in C of degree ml − d So |fl (z)|ml = c1 f (z) ml −d ˆ in+1 (f )(z)|}, max{|Qˆ i1 (f )(z)|, , |Q where c1 is a positive constant and depends only on the coefficients of bjl , ≤ i ≤ n + 1, ≤ l ≤ k, thus depends only on the coefficients of Qij , ≤ j ≤ n + Therefore, f (z) d ˆ i1 (f )(z)|, , |Qˆ in+1 (f )(z)|} ≤ c1 max{|Q (4) Since Hˆ , , Hˆ q are in n-subgeneral position in Pk (C), there exists an injective map α : {0, , k} → {i1 , , in+1 } such that Hˆ α(0), , Hˆ α(k) are linearly independent Hence, ∩kl=0 Dα(l) ∩ Pk (C) = ∅ By (3), (4), and Lemma 2.3, we have q j =1 f (z) d Qˆ j |Qˆ j (f )(z)| ωj n+1 ≤c j =1 k+1 ≤c l=0 f (z) d Qˆ ij ˆ ij (f )(z)| |Q ωij f (z) d Qˆ α(l) , ˆ α(l)(f )(z)| |Q (5) 226 Do Duc Thai and Ninh Van Thu where c > is constant, depending only on the coefficients of Qˆ ij , ≤ j ≤ n + Therefore, q ωj log j =1 f (z) d Qˆ j |Qˆ j (f )(z)| f (z) d Qˆ α(l) |Qˆ α(l) (f )(z)| k ≤ log l=0 f (z) d Qˆ α(l) ˆ α(l)(f )(z)| |Q k ≤ max log α∈T + O(1) l=0 + O(1), (6) where T is the set of all injective maps α : {0, , k} → {1, , q} such that ∩kl=0 Supp(Dα(l) ) ∩ Pk (C) = ∅ Thus, q log j =1 q f (reiθ ) d Qˆ j = (1 − ˆ j (f )(reiθ )| |Q j =1 ωj ) log f (reiθ ) d Qˆ j |Qˆ j (f )(reiθ )| q + ωj log j =1 ≤ d[2n − k + − q − f (reiθ ) d Qˆ j |Qˆ j (f )(reiθ )| (k + 1)] log f (reiθ ) ωj ] log |Qˆ j (f )(reiθ )| [1 − j =1 k + log max α∈T l=0 f (reiθ ) d Qˆ α(l) + O(1) |Qˆ α(l) (f )(reiθ )| (7) We now use Ru’s argument (see the proof of Main Theorem in [9]) to estimate k log max α∈T l=0 f (reiθ ) d Qˆ α(l) ˆ α(l)(f )(reiθ )| |Q ˆ (x) : · · · : Qˆ q (x)] ∈ Pq−1 (C) and let Y = φ(Pk (C)) By Define a map φ : x ∈ Pk (C) → [Q the ‘in n-subgeneral position’ assumption, φ is a finite morphism on Pk (C) and Y is a complex projective subvariety of Pq−1 (C) We also have that dim Y = n For every a q a a = (a1 , , aq ) ∈ Z≥0 , denote by y a = y1 · · · yq q Let m be a positive integer Put nm := HY (m) − 1, qm := q +m−1 − 1, m where HY is the Hilbert function of Y Consider the Veronese embedding φm : Pq−1 (C) → Pqm (C) : y → [y a0 : · · · : y aqm ], where y a0 , , y aqm are the monomials of degree m in y1 , , yq , in some order Denote by Ym the smallest linear subvariety of Pqm (C) containing φm (Y ) It is known that Ym is an nm -dimensional linear subspace of Pqm (C), where nm = HY (m) − Since Ym is an nm -dimensional linear subspace of Pqm (C), there are linear forms L0 , , Lqm ∈ C[w0 , , wnm ] such that the map ψm : w ∈ Pqm (C) → [L0 : · · · : Lqm ] ∈ Ym The Second Main Theorem 227 −1 ◦ φ is an injective map from Y is a linear isomorphism from Pqm (C) to Ym Thus ψm m n k −1 ◦ φ ◦ m into P (C) Let f : C → P (C) be the given holomorphic map and let F = ψm m n m φ ◦ f : C → P (C) Then F is a holomorphic map Furthermore, since f is algebraically nondegenerate, F is linearly nondegenerate For > given in Theorem 1.1, we have the following estimate (see inequality (3.19) of [9]): f (reiθ ) d Qˆ α(l) |Qˆ α(l) (f (reiθ ))| k max log α∈T l=0 nm ≤ 3mNd max log β∈K t =0 + d(k + 1) + 3N F˜ (reiθ ) Lβ(t ) − (nm + 1) log F˜ (reiθ ) |Lβ(t )(F˜ (reiθ ))| log f (reiθ ) + O(1), (8) where N := (2n − k + 1)(k + 1) and F˜ is a reduced representation of F Therefore, we get f (reiθ ) d Qˆ j |Qˆ j (f )(reiθ )| q log j =1 q ≤ d(2n − k + 1) log f (reiθ ) − [1 − ˆ j (f )(reiθ )| ωj ] log |Q j =1 nm + 3mNd + 3N max log β∈K t =0 log f (reiθ ) + O(1) nm 2π max log β∈K (9) = to a holomorphic map F and linear forms L0 , , Lqm , Applying Theorem 2.1 with we obtain that F˜ (reiθ ) Lβ(t ) − (nm + 1) log F˜ (reiθ ) iθ ˜ |Lβ(t ) (F (re ))| t =0 F˜ (reiθ ) Lβ(t ) − (nm + 1) log F˜ (reiθ ) iθ ˜ |Lβ(t ) (F (re ))| dθ 2π ≤ TF (r) ≤ dmT f (r) Since ≤ − (10) ωj < 1, (n + 1)/(k + 1) ≤ 2π ≤ (2n − k + 1)(k + 1), ˆ j (f )(reiθ )| dθ > O(1) log |Q 2π and by (10), after taking integration on both sides of (9) we get q mf (r, Dj ) ≤ (d(2n − k + 1) + /3)Tf (r) + C, j =1 where C is a constant, independent of r Take r big enough so we can make C ≤ /3Tf (r) 228 Do Duc Thai and Ninh Van Thu Hence, we have q mf (r, Dj ) ≤ (d(2n − k + 1) + )Tf (r) j =1 ✷ The proof is complete Second Main Theorem with truncated counting functions for the nondegenerate case In this section we recall Corvaja and Zannier’s filtration as made more explicit in [3] Details of proofs can be found in [3], and also [2] For a fixed positive integer N, denote by VN the space of homogeneous polynomials of degree N in C[x0 , , xn ] Let γ1 , , γn be homogeneous polynomials in L EMMA 3.1 (See [2, Lemma 5]) C[x0 , , xn ] and assume that they define a subvariety of Pn (C) of dimension Then for all N ≥ nj=1 deg γj , dim VN = deg γ1 · · · deg γn (γ1 , , γn ) ∩ VN Throughout the rest of this paper, we shall use the lexicographic ordering on n-tuples (i1 , , in ) ∈ Nn of natural numbers Namely, (j1 , , jn ) > (i1 , , in ) if and only if for some b ∈ {1, , n} we have jl = il for l < b and jb > ib Given an n-tuple (i) = (i1 , , in ) of non-negative integers, we denote σ (i) := j ij Let γ1 , , γn ∈ C[x0, , xn ] be the homogeneous polynomials of degree d that define a zero-dimensional subvariety of Pn (C) We now recall Corvaja and Zannier’s filtration of VN Arrange, by the lexicographic order, the n-tuples (i) = (i1 , , in ) of non-negative integers such that σ (i) ≤ N/d Define the spaces W(i) = WN,(i) by γ1e1 · · · γnen VN−dσ (e) W(i) = (e)≥(i) Clearly, W(0, ,0) = VN and W(i ) ⊂ W(i) if (i ) > (i), so the W(i) is a filtration of VN Next, we recall some results about the quotients of consecutive spaces in the filtration L EMMA 3.2 (See [4, Proposition 3.3]) For any non-negative integer k, the dimension of the vector space Vk /(γ1 , , γn ) ∩ Vk is equal to the number of n-tuples (i1 , , in ) ∈ Nn such that i1 + · · · + in ≤ k and ≤ i1 , , in ≤ d − In particular, for all k ≥ n(d − 1), we have Vk = d n dim (γ1 , , γn ) ∩ Vk L EMMA 3.3 (See [2, Lemma 6]) There is an isomorphism W(i) /W(i ) ∼ = VN−dσ (i) (γ1 , , γn ) ∩ VN−dσ (i) Furthermore, we may choose a basis of W(i) /W(i ) from the set containing all equivalence classes of the form γ1i1 · · · γnin q modulo W(i ) with q being a monomial in x0 , , xn with total degree N − dσ (i) The Second Main Theorem 229 We now combine Lemma 3.2 and Lemma 3.3 to explicitly calculate the dimension of the quotient spaces, which we denote by (i) L EMMA 3.4 If σ (i) ≤ N + n/d − n then Let = (i) following lemma (i) ij = (1/n) (i) := dim W(i) /W(i ) = d n and let M = (i) σ (i) (i) (i) (i) We have the L EMMA 3.5 For any k1 , , kn ∈ N, we have k1 + · · · + kn − n ij kj − M (i) + M2 ≤ , j =1 (i) where x + = max{x, 0} Proof For given k1 , , kn ∈ N, we have n n j =1 (i) n ij kj = (i) kj j =1 (i) ij = kj j =1 (i) Therefore, k1 + · · · + kn − n (i) n (i) j =1 (i) + j =1 M ij kj , M j =1 (i) ≤ ij kj − M n = n ij kj − (i) + j =1 (i) = ij kj − M (i) = M2 (11) (i) ✷ This completes the proof Proof of Theorem 1.2 Given z ∈ C such that f (z) ∈ Pn (C), there exists a renumbering {i1 , , iq } of the indices {1, , q} such that |Qi1 ◦ f (z)| ≤ |Qi2 ◦ f (z)| ≤ · · · ≤ |Qiq ◦ f (z)| (12) Since Q1 , , Qq are in general position and by Hilbert’s Nullstellensatz, for any integer l, ≤ l ≤ n, there is an integer ml ≥ d such that n+1 m xl l = bjl (x0 , , xn )Qij (x0 , , xn ), j =1 where bjl , ≤ j ≤ n + 1, ≤ l ≤ n, are the homogeneous forms with coefficients in C of degree ml − d So |fl (z)|ml = c1 f (z) ml −d max{|Qi1 (f )(z)|, , |Qin+1 (f )(z)|}, 230 Do Duc Thai and Ninh Van Thu where c1 is a positive constant and depends only on the coefficients of bil , ≤ i ≤ n + 1, ≤ l ≤ n, thus depends only on the coefficients of Qi , ≤ i ≤ n + Therefore, d f (z) ≤ c1 max{|Qi1 (f )(z)|, , |Qin+1 (f )(z)|} (13) By (12) and (13), we have q j =1 n f (z) d Qij f (z) d ≤c |Qj (f )(z)| |Qij (f )(z) j =1 (14) Hence, by the definition, we get q max {i1 ,i2 , ,in } j =1 n 2π mf (r, Qj ) ≤ ln j =1 f (reiθ ) d dθ + O(1) Qij (f )(reiθ ) 2π (15) For n distinct polynomials r1 , , rn , we have constructed a filtration W(i) of VN Set M = MN := dim VN We now choose a suitable basis {ψ1 , , ψM } for VN in the following way We start with the last nonzero W(i) and pick any basis of it Then we continue inductively as follows Suppose (i ) > (i) are consecutive n-tuples such that dσ (i), dσ (i ) ≤ N and assume that we have chosen a basis of W(i ) It follows directly from the definition that we may pick representatives in W(i) for the quotient space W(i) /W(i ) , of the form r1i1 r2i2 · · · rnin q, where q ∈ VN−dσ (i) We extend the previously constructed basis in W(i ) by adding these representatives In particular, we have obtained a basis for W(i) and our inductive procedure may go on unless W(i) = VN , in which case we stop In this way, we obtain a basis {ψ1 , , ψM } for VN Now let φ1 , , φM be a fixed basis of VN Then {ψ1 , , ψM } can be written as linear forms L1 , , LM in φ1 , , φM so that ψt (f ) = Lt (F ), where F = (φ1 (f ) : · · · : φM (f )) The linear forms L1 , , LM are linearly independent, and we know, from the assumption of algebraic nondegeneracy of f , that F is linearly nondegenerate M We now estimate ln M t =1 |Lt (F )(z)| = ln t =1 |ψt (F )(z)| Let ψ be an element of the i i basis, constructed with respect to W(i) /W(i ) So we may write ψ = r11 r22 · · · rnin q, where q ∈ VN−dσ (i) Then we have a bound |ψ(f )(z)| ≤ |r1 (f )(z)|i1 · · · |rn (f )(z)|in |q(f )(z)| ≤ c2 |r1 (f )(z)|i1 · · · |rn (f )(z)|in f (z) N−dσ (i) , where c2 is positive constant depending only on ψ, not on f and z Observe that there are precisely (i) such functions ψ in our basis Hence, M |ψt (F )(z)| ≤ ln t =1 (i) (i1 ln |r1 (f )(z)| + · · · + in ln |rn (f )(z)|) (i) + ln f (z) (i) (N (i) − dσ (i)) + c3 , The Second Main Theorem 231 where c3 depends only on ψ, not on f and z Here the summation is taken over the n-tuples with σ (i) ≤ N/d Therefore, n ln j =1 M f (z) d F (z) ln ≤ − M ln F (z) |rj (f )(z)| |L (F )(z)| t t =1 + q mf (r, Qj ) ≤ NM ln f (z) + 2π max ln K j =1 + c3 NM j ∈K (16) , dθ F (reiθ ) − MTF (r) iθ |Lj (F )(re )| 2π Tf (r) + O(1) (17) By Theorem 2.2, we obtain q dqT f (r) ≤ Nf (r, Qj ) − NW (r, 0) + NM Tf (r) + O(ln Tf (r)), (18) j =1 where W is the Wronskian of F1 , , FM q We now estimate j =1 Nf (r, Qj ) − (1/ )NW (r, 0) on the right-hand side of (18) For each z ∈ C, without loss of generality, we assume that Qj (f ) vanishes at z for ≤ j ≤ q1 and Qj (f ) does not vanish at z for j > q1 Since D1 , D2 , , Dq are in general position, q1 ≤ n There are integers kj ≥ and nowhere vanishing holomorphic functions gj in a neighborhood U of z such that Qj (f ) = (ζ − z)kj gj , for j = 1, , q, where kj = if q1 < j ≤ q For {Q1 , , Qn } ⊂ {Q1 , , Qq }, we can obtain a basis {ψ1 , , ψM } of VN and linearly independent linear forms L1 , , LM such that ψt (f ) = Lt (F ) By the property of the Wronskian, W = W (F1 , , FM ) = CW (L1 (F ), , LM (F )) =C ψ1 (f ) (ψ1 (f )) ψ2 (f ) (ψ2 (f )) ψM (f ) (ψM (f )) (ψ1 (f ))(M−1) (ψ2 (f ))(M−1) (ψM (f ))(M−1) (19) Let ψ be an element of a basis, constructed with respect to W(i) /W(i ) So we may write ψ = Qi11 · · · Qinn q, q ∈ VN−dσ (i) We have ψ(f ) = (Q1 (f ))i1 · · · (Qn (f ))in q(f ), where Qij (f ) = (ζ − z)kj g, j = 1, , n Thus, (ζ − z)( W (L1 (F ), , LM (F )) = n + j=1 ij kj −M) h(ζ ), where h(ζ ) is a holomorphic function defined on U Thus W vanishes at z at least n + (i) (i) ( j =1 ij kj − M) By Lemma 3.5, we have k1 + · · · + kn − n ij kj − M (i) (i) j =1 + ≤ M2 232 Do Duc Thai and Ninh Van Thu (ν) This, together with definitions of Nf (r, Qj ), NW (r, 0), and ND , implies that q Nf (r, Qj ) − j =1 ([M / ]) NW (r, 0) ≤ Nf (r, D), where D = D1 ∪ · · · ∪ Dq and N (ν) (r, div(f, D)) is the counting function with the q truncation level ν of the divisor div(f, D) defined by div(f, D) = j =1 (d/dj ) div(f, Dj ) The proof of Theorem 1.2 is now complete ✷ Recall that = (i) (i) ij = (1/n) (i) (i) σ (i) and M = (i) (i) L EMMA 3.6 If ≤ x ≤ 2/n(n + 1)d , then n(n + 1) dx (1 − x)(1 − 2x) · · · (1 − nx) ≤1+ (1 − dx)(1 − dx) · · · (1 − n dx) (20) Proof The proof is by induction on n For n = 1, this inequality is valid Assume that the inequality holds for n − 1, i.e n(n − 1) dx (1 − x)(1 − 2x) · · · (1 − (n − 1)x) ≤1+ (1 − dx)(1 − dx) · · · (1 − (n − 1) dx) Then, since −nx ≤ −n2 (n + 1)d x /2, we get dn(n − 1)x − nx (1 − x)(1 − 2x) · · · (1 − nx) ≤ 1+ (1 − dx)(1 − dx) · · · (1 − n dx) − n dx ≤ + n(n − 1) dx/2 − nx − n dx + n(n + 1) dx/2 − n dx − n2 (n + 1) d x /2 − n dx n(n + 1) dx (21) ≤1+ ≤ ✷ This completes the inductive proof Proof of Corollary 1.3 For N > dn, by Lemma 3.4 we have the estimate := ij (i) ≥ σ (i)≤N/d = Moreover, M= ij (i) σ (i)≤(N+n)/d−n dn n+1 n+1 ij i1 +···+in+1 =(N+n)/d−n j =1 = d n (N + n)/d n+1 n = (N + n)(N + n − d) · · · (N + n − nd) d(n + 1)! n+N n = N +n −n d (N + 1)(N + 2) · · · (N + n) n! The Second Main Theorem 233 Therefore, by Lemma 3.6, we get NM ≤ d(n + 1) = d(n + 1) N(N + 1)(N + 2) · · · (N + n) (N + n)(N + n − d) · · · (N + n − nd) (1 − (1 − ≤ d(n + 1) + = d(n + 1) + n N+n )(1 − N+n ) · · · (1 − N+n ) d 2d nd N+n )(1 − N+n ) · · · (1 − N+n ) dn(n + 1) N +n d n(n + 1)2 N +n (22) for N ≥ n(n + 1)d /2 − n For < ≤ (n + 1)/d, take N = [dn(n + 1)2 /2 − n] Then N ≥ n(n + 1)d /2 − n Thus, by (22) we obtain NM/ ≤ d(n + + ) Hence, M 2/ (N + 1)(N + 2) · · · (N + n) NM M ≤ d(n + + ) N Nn! n−1 n(n + 1) N exp ≤ d(n + + ) n! 2N = ≤ d(n + + ) ≤ d(n + + n ≤ 2den ( −1 dn(n + 1)2 /2 − n)n−1 )en n! d(n + 1)2 −1 d(n + 1)2 −1 n−1 n−1 , where the last two inequalities are implied by nn /n! ≤ en The proof is complete ✷ A unicity theorem for algebraically nondegenerate curves into P2 (C) First of all, we give the SMT with explicit truncated counting functions for nondegenerate holomorphic curves into P2 (C) and hypersurface targets located in general position Let Qj , ≤ j ≤ q, be the homogeneous polynomials of degree dj , located in general d/d position in P2 (C) Replacing Qj by Qj j if necessary, where d is the least common multiple of the dj , we may assume that Q1 , , Qq have the same degree of d By Lemma 3.2 and Lemma 3.3, for each i = (i1 , i2 ) ∈ N2 with σ (i) = i1 + i2 ≤ N/d, we have (i) is equal to the number of 2-tuples (j1 , j2 ) ∈ N2 such that j1 + j2 ≤ N − dσ (i) and ≤ j1 , j2 ≤ d − Take N = 2d By a computation, we get ⎧ ⎨1 (i) = d + 3d − ⎩ if σ (i) = 2, if σ (i) = 234 Do Duc Thai and Ninh Van Thu Thus, = MN d + 3d + , = M= 2d + = (d + 1)(2d + 1), 4d(d + 1)(2d + 1) , d + 3d + M2 = 2(d + 1)2 (2d + 1)2 d + 3d + Theorem 1.2 in this case is now stated as follows C OROLLARY 4.1 Let f : C → P2 (C) be an algebraically nondegenerate holomorphic q curve, and let {Dj }j =1 be hypersurfaces in P2 (C) of degree dj in general position Let d be the least common multiple of the dj Then d q− 4(d + 1)(2d + 1) Tf (r) ≤ N d + 3d + where div(f, D) = 2(d+1)2 (2d+1)2 d +3d+4 (r, div(f, D)) + O(ln Tf (r)), q j =1 (d/dj ) div(f, Dj ) L EMMA 4.2 (See [5, Lemma 5.1]) Let A1 , , Ak be pure (m − 1)- dimensional analytic subsets of Cm with codim(Ai ∩ Aj ) ≥ whenever i = j Let f1 , f2 be linearly nondegenerate holomorphic mappings of C into Pn (C) Then there exists a dense subset P ⊂ Cn+1 such that, for any p := (p0 , , pn ) ∈ P, the hyperplane Hp defined by p0 w0 + ∗ · · · + pn wn = satisfies codim{∪kj =1 Aj ∩ fi−1 (Hp )} ≥ 2, i ∈ {1, 2} Proof of Theorem 1.4 Assume that f ≡ g Then there exist hyperplanes H1 , H2 in P2 (C) such that f −1 (Qj ) ∩ f −1 (Hi ) = ∅, g −1 (Qj ) ∩ g −1 (Hi ) = ∅ for all i ∈ {1, 2} and (f, H1 ) (g, H1 ) ≡ (f, H2 ) (g, H2 ) Indeed, suppose that this assertion does not hold Then by Lemma 4.2, we have (f, H1 ) (g, H1 ) ≡ (f, H2 ) (g, H2 ) for almost hyperplanes H1 , H2 in P2 (C) This is impossible unless f ≡ g This is a contradiction By the assumption of Theorem 1.4 and by the First Main Theorem we have q (1) j =1 Nf (r, Qj ) ≤ N (f,H1 ) − (g,H1 ) (r) (f,H2 ) (g,H2 ) ≤ T (f,H1 ) − (g,H1 ) (r) + O(1) (f,H2 ) (g,H2 ) ≤ T (f,H1 ) (r) + T (g,H1 ) (r) + O(1) (f,H2 ) (g,H2 ) ≤ Tf (r) + Tg (r) + O(1) The Second Main Theorem 235 Similarly, q Ng(1) (r, Qj ) ≤ Tf (r) + Tg (r) + O(1) j =1 Thus, q j =1 (Nf(1) (r, Qj ) + Ng(1) (r, Qj )) ≤ Tf (r) + Tg (r) + O(1) (23) By Corollary 4.1, d q− ≤N 4(d + 1)(2d + 1) Tf (r) d + 3d + (2d+1)2 ]) d +3d+4 ([ 2(d+1) (r, div(f, D)) + o(Tf (r)) < 2(d + 1)2 (2d + 1)2 + N (1) (r, div(f, D)) + o(Tf (r)) d + 3d + < 2(d + 1)2 (2d + 1)2 +1 d + 3d + (24) (25) q (1) j =1 Nf (r, Qj ) + o(Tf (r)) (26) Similarly, (d(q − < 4(d + 1)(2d + 1) )Tg (r) d + 3d + 2(d + 1)2 (2d + 1)2 +1 d + 3d + (27) q Ng(1) (r, Qj ) + o(Tg (r)) (28) j =1 By (23), (26), and (28) we get d q− < 4(d + 1)(2d + 1) (Tf (r) + Tg (r)) d + 3d + 4(d + 1)2 (2d + 1)2 + (Tf (r) + Tg (r)) + o(Tf (r) + Tg (r)) d + 3d + This is a contradiction, since q≥ The proof is completed 4(d + 1)(2d + 1) 4(d + 1)2 (2d + 1)2 + + d d(d + 3d + 4) d + 3d + ✷ Acknowledgements We would like to thank Professor Yum-Tong Siu for his helpful suggestions in proving Lemma 2.4 The research of the authors is partially supported by a NAFOSTED grant of Vietnam (Grant No 101.01.38.09) and the research of the second author is partially supported by a Research grant of the College of Science, Vietnam National University (Grant No TN10-04) 236 Do Duc Thai and Ninh Van Thu R EFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] T T H An and H T Phuong An explicit estimate on multiplicity truncation in the Second Main Theorem for holomorphic curves encountering hypersurfaces in general position in projective space Houston J Math 35 (2009), 775–786 T T H An and J T.-Y Wang An effective Schmidts subspace theorem for non-linear forms over function fields J Number Theory 125 (2007), 210–228 P Corvaja and U M Zannier On a general Thues equation Amer J Math 126 (2004), 1033–1055 G Dethloff and T V Tan A second main theorem for moving hypersurfaces targets Houston J 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need the following general form of the SMT for holomorphic curves intersecting hyperplanes They are stated and proved... 1) + )Tf (r) j =1 ✷ The proof is complete Second Main Theorem with truncated counting functions for the nondegenerate case In this section we recall Corvaja and Zannier’s filtration as made more... 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