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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 \((2,3)\).

MSC:
14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
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[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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