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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
##### Keywords:
torus curve; Newton boundary
Full Text:
##### References:
 [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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