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Classification of local singularities on torus curves of type \((2,5)\). (English) Zbl 1197.14025
The paper deals with a special class of degree \(10\) reduced plane curves \(C \subset \mathbb{P}^2\), those of torus type \((2,5)\).
A complex projective plane curve \(C\) defined by the equation \(f(x,y)=0\) in the affine chart \(\mathbb{C}^2:= \mathbb{P}^2- \{Z=0\}\), is called torus curve of type \((p,q)\) if one can write \(f={f^q}_p+ {f^p}_q\), for some polynomials \(f_p, f_q\) of degree \(p\) and \(q\) respectively.
The author gives a topological classification of the local singularity of the curve \(C: {f^2}_5+{f^5}_2=0\) at the origin, in the case that the origin is an isolated singularity and the curves \(C_2: f_2=0\) and \(C_5: f_5=0\) pass through it (inner singularity). To obtain this result , he follows the method used by D. T. Pho in [Kodai Math. J. 24 (2), 259–284 (2000; Zbl 1072.14031)], where the classification of local and global configurations of singularities is given for sextics of torus type \((2,3)\).

14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
Full Text: DOI
[1] B. Audoubert, T. C. Nguyen and M. Oka, On Alexander polynomials of torus curves, J. Math. Soc. Japan, 57 (4) (2005), 935-937. · Zbl 1085.14025
[2] M. Oka, On the bifurcation of the multiplicity and topology of the Newton boundary, J. Math. Soc. Japan, 31 (1979), 435-450. · Zbl 0408.35012
[3] M. Oka, On the resolution of the hypersurface singularities, in, Complex Analytic Singularities , volume 8 of Adv. Stud. Pure Math., North-Holland, Amsterdam (1987), 405-436. · Zbl 0622.14012
[4] M. Oka, Non-degenerate complete intersection singularity Hermann, Paris 1997. · Zbl 0930.14034
[5] M. Oka, Geometry of cuspidal sextics and their dual curves, in, Singularities–Sapporo 1998 , volume 29 of Adv. Stud. Pure Math., Kinokuniya, Tokyo (2000), 245-277. · Zbl 1020.14008
[6] D. T. Pho, Classification of singularities on torus curves of type \((2,3)\), Kodai Math. J., 24 (2) (2001), 259-284. · Zbl 1072.14031
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