Classification of local singularities on torus curves of type \((2,5)\). (English) Zbl 1197.14025

Kawashima, Masayuki

Tokyo J. Math. 32, No. 2, 313-348 (2009).

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 \((2,3)\).

Reviewer: Caterina Cumino (Torino)

MSC:

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

Keywords:

torus curve; Newton boundary

Citations:

Zbl 1072.14031

References:

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