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


Author: Kiyoshi Niinō
Journal: Proc. Amer. Math. Soc. 59 (1976), 81-88
MSC: Primary 32H25; Secondary 30A70
DOI: https://doi.org/10.1090/S0002-9939-1976-0414943-2
MathSciNet review: 0414943
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0414943-2
Keywords: Holomorphic curve, associated curve of rank $ k$, projective space, order function, defect (deficiency), general position, degenerate curve
Article copyright: © Copyright 1976 American Mathematical Society

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