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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Rational curve with one cusp. II


Author: Hisao Yoshihara
Journal: Proc. Amer. Math. Soc. 100 (1987), 405-406
MSC: Primary 14H20; Secondary 14H45
DOI: https://doi.org/10.1090/S0002-9939-1987-0891134-X
MathSciNet review: 891134
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Abstract: Let $ C$ be a plane curve with $ C - \{ P\} \cong {{\mathbf{A}}^1}$ for some point $ P \in C$ and degree $ d \geq 3$. Let $ \{ {e_1}, \ldots ,{e_t}\} $ be the multiplicities of the infinitely near singular points of $ P$. Then the following three conditions are equivalent:

(1) $ C\backslash L \cong {{\mathbf{A}}^1}$ for some line $ L$,

(2) $ d = {e_1} + {e_2}\, ({\text{in}}\;{\text{case}}\;t = 1,\;{\text{let}}\;{e_2} = 1)$,

(3) $ R = {d^2} - \sum\nolimits_{i = 1}^t {e_i^2 - {e_t} + 1 \geq 3} $.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0891134-X
Article copyright: © Copyright 1987 American Mathematical Society