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Classification of sextics of torus type. (English) Zbl 1062.14036

From the introduction: We consider an irreducible sextic \(C\) of torus type defined by \(\{(X,Y,Z)\in\mathbb{P}^2;\;F_2(X,Y,Z)^3+ F_3(X,Y, Z)^2=0\}\). We say that \(C\) is tame if its singularities are contained in \(V(F_2)\cap V(F_3)\). For tame sextics of torus type, there are 25 local singularity types among which 20 appear on irreducible sextics of torus type [D. T. Pho, Kodai Math. J. 24 (2), 259–284 (2000; Zbl 1072.14031)]. As global singularities, there are 43 configurations of singularities on irreducible tame torus curves. In this paper, we complete the classification of configurations of the singularities on irreducible sextics of torus type. We show that there exist 121 configurations and there are 5 pairs and a triple of configurations for which the corresponding moduli spaces coincide, ignoring the respective torus decomposition.

MSC:

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

Citations:

Zbl 1072.14031
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References:

[1] E. Horikawa, On deformations of quintic surfaces. Invent. Math., 31 (1975), 43-85. · Zbl 0317.14018
[2] J. Milnor, Singular Points of Complex Hypersurface , Annals Math. Studies 61 (1968), Princeton Univ. Press. · Zbl 0184.48405
[3] M. Namba, Geometry of Projective Algebraic Curves , Decker (1984). · Zbl 0556.14012
[4] M. Oka, Geometry of reduced sextics of torus type, preprint in, preparation. · Zbl 1047.14002
[5] M. Oka, Geometry of cuspidal sextics and their dual curves, Singularities–Sapporo 1998 , Kinokuniya (2000), 245-277. · Zbl 1020.14008
[6] M. Oka and D. T. Pho, Fundamental group of sextic of torus type, Trends in Singularities , Birkhäuser (2002), 151-180. · Zbl 1054.14022
[7] D. T. Pho, Classification of singularities on torus curves of type \((2,3)\), Kodai Math. J., 24 (2) (2000), 259-284. · Zbl 1072.14031
[8] T. Shioda and H. Inose, On singular \(K3\) surfaces, Complex analysis and algebraic geometry , Iwanami Shoten (1977), 119-136. · Zbl 0374.14006
[9] H.-o. Tokunaga, (2,3) torus sextics and the Albanese images of 6-fold cyclic multiple planes, Kodai Math. J., 22 (2) (1999), 222-242. · Zbl 0990.14010
[10] J.-G. Yang, Sextic curves with simple singularities, Tohoku Math. J., 48 (2) (1996), 203-227. · Zbl 0866.14014
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