Rational curve with one cusp. II
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- by Hisao Yoshihara PDF
- Proc. Amer. Math. Soc. 100 (1987), 405-406 Request permission
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) $R = {d^2} - \sum \nolimits _{i = 1}^t {e_i^2 - {e_t} + 1 \geq 3}$.References
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M. H. Gizatullin, On affine surfaces that can be completed by a nonsingular rational curve, Math. USSR-Izv. 4 (1970), 787-810.
- Hisao Yoshihara, Rational curve with one cusp, Proc. Amer. Math. Soc. 89 (1983), no. 1, 24–26. MR 706501, DOI 10.1090/S0002-9939-1983-0706501-0
- Hisao Yoshihara, On open algebraic surfaces $\textbf {P}^{2}-C$, Math. Ann. 268 (1984), no. 1, 43–57. MR 744327, DOI 10.1007/BF01463872
Additional Information
- © Copyright 1987 American Mathematical Society
- 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