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

Abstract:

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