Deficiencies of the associated curves of a holomorphic curve in the projective space

Abstract

Let k x _kx be the nonconstant associated holomorphic curve of rank k ( 1 ≦ k ≦ n − 1 ) k(1 \leqq k \leqq n - 1) of a transcendental holomorphic curve x : C → P n C x:{\mathbf {C}} \to {P_n}{\mathbf {C}} . It is proved that if 1 ≦ k ≦ n − 2 1 \leqq k \leqq n - 2 and A j k ∈ P l ( k ) − 1 C , j = 1 , … , 2 l ( k ) − 2 ( l ( k ) = ( k + 1 n + 1 ) ) A_j^k \in {P_{l(k) - 1}}{\mathbf {C}},j = 1, \ldots ,2l(k) - 2(l(k) = (_{k + 1}^{n + 1})) are in general position and ⟨ k x , A j k ⟩ ≢ 0 {\langle _k}x,A_j^k\rangle \not \equiv 0 for all A j k A_j^k , then ∑ j = 1 2 l ( k ) − 2 δ k ( A j k ) ≦ 2 l ( k ) − 3 \sum \nolimits _{j = 1}^{2l(k) - 2} {{\delta _k}(A_j^k) \leqq 2l(k) - 3} and that in the case when k = n − 1 , ∑ A n − 1 δ n − 1 ( A n − 1 ) ≦ l ( n − 1 ) k = n - 1,\sum \nolimits _{{A^{n - 1}}} {{\delta _{n - 1}}({A^{n - 1}}) \leqq l(n - 1)} , where { A n − 1 } \{ {A^{n - 1}}\} is a finite subset of P l ( n − 1 ) − 1 C {P_{l(n - 1) - 1}}{\mathbf {C}} in general position such that ⟨ n − 1 x , A n − 1 ⟩ ≢ 0 {\langle _{n - 1}}x,{A^{n - 1}}\rangle \not \equiv 0 for all A n − 1 {A^{n - 1}} . These are sharp.