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\(\pi_1\)-equivalent weak Zariski pairs. (English) Zbl 1095.14025

In the moduli space \({\mathcal M}(\Sigma, d)\) of reduced curves in \({\mathbb C\mathbb P}^2\) with given degree \(d\) and prescribed configuration of singularities \(\Sigma\), a couple of reduced curves \(C\) and \(C'\) is said to be a weak Zariski pair if the pairs of spaces \(({\mathbb C\mathbb P}^2, C)\) and \(({\mathbb C\mathbb P}^2, C')\) are not homeomorphic; a weak Zariski \(k\)-ple is a \(k\)-ple \((D_1, \dots, D_k)\) of curves in \({\mathcal M}(\Sigma, d)\) such that the pairs of spaces \(({\mathbb C\mathbb P}^2, D_i)\) and \(({\mathbb C\mathbb P}^2, D_j)\) are not homeomorphic [M. Oka, Tokyo J. Math. 26, No. 2, 301–327 (2003; Zbl 1047.14002)]. The first example of such pairs appears in the works by O. Zariski, then several examples were found by many authors. In this paper the first example is given of a weak Zariski pair \(C,C'\) such that the fundamental groups \(\pi_1({\mathbb P}^2- C)\) and \(\pi_1({\mathbb P}^2- C')\) are isomorphic. The curves \(C\) and \(C'\) in the example are sextics with configuration of singularities \(\{D_{10}+A_5+A_4\}\). As an application, the authors present a new weak Zariski 4-ple.

MSC:

14H20 Singularities of curves, local rings
14H30 Coverings of curves, fundamental group

Citations:

Zbl 1047.14002
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References:

[1] E. Artal Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), 223-247. · Zbl 0823.14013
[2] E. Artal Bartolo and J. Carmona Ruber, Zariski pairs, fundamental groups and Alexander polynomials, J. Math. Soc. Japan 50 (1998), 521-543. · Zbl 0904.14011 · doi:10.2969/jmsj/05030521
[3] E. Brieskorn and H. Knörrer, Plane algebraic curves , Birkhäuser (1986). · Zbl 0588.14019
[4] D. Chéniot, Une démonstration du théorème de Zariski sur les sections hyperplanes d’une hypersurface projective et du théorème de van Kampen sur le groupe fondamental du complémentaire d’une courbe projective plane, Compositio Math. 27 (1973), 141-158. · Zbl 0294.14010
[5] A. Dimca, Singularities and topology of hypersurfaces , Springer-Verlag, New-York (1992). · Zbl 0753.57001
[6] G.M. Greuel, C. Lossen and E. Shustin, Castelnuovo function, zero-dimensional schemes and singular plane curves, J. Algebraic Geom. 9 (2000), 663-710. · Zbl 1037.14009
[7] J. Harris, On the Severi problem, Invent. Math. 84 (1986), 345-461. · Zbl 0596.14017 · doi:10.1007/BF01388741
[8] Lê D.T. and C. P. Ramanujam, The invariance of Milnor number implies the invariance of the topological type, Amer. J. Math. 98 (1976), 67-78. · Zbl 0351.32009 · doi:10.2307/2373614
[9] J. Milnor, Singular points of complex hypersurfaces , Annals Math. Studies 61 , Princeton Univ. Press (1968). · Zbl 0184.48405
[10] M. Oka, Symmetric plane curves with nodes and cusps, J. Math. Soc. Japan 44 (1992), 375-414. · Zbl 0767.14011 · doi:10.2969/jmsj/04430375
[11] M. Oka, Two transforms of plane curves and their fundamental groups, J. Math. Sci. Univ. Tokyo 3 (1996), 399-443. · Zbl 0935.14013
[12] M. Oka, Geometry of cuspidal sextics and their dual curves, Singularities - Sapporo 1998 Advanced Studies in Pure Math. 29 (2000), 247-277.
[13] M. Oka, Flex curves and their applications, Geometriae Dedicata 75 (1999), 67-100. · Zbl 0952.14020 · doi:10.1023/A:1005004123844
[14] M. Oka, A survey on Alexander polynomials of plane curves, Séminaires et Congrès No. 10 (2005), 209-232. · Zbl 1093.14037
[15] M. Oka, A new Alexander-equivalent Zariski pair, Acta Math. Vietnam. 27 (3) (2002), 349-357. · Zbl 1061.14023
[16] M. Oka, Geometry of reduced sextics of torus type, Tokyo J. Math. 26 No. 2 (2003), 301-327. · Zbl 1047.14002 · doi:10.3836/tjm/1244208593
[17] M. Oka and D. T. Pho, Classification of sextics of torus type, Tokyo J. Math. 25 (2) (2002), 399-433. · Zbl 1062.14036 · doi:10.3836/tjm/1244208862
[18] D. T. Pho, Classification of singularities on torus curves of type (2,3), Kodai Math. J. 24 (2001), 259-284. · Zbl 1072.14031 · doi:10.2996/kmj/1106168786
[19] I. Shimada, A note on Zariski pairs, Compositio Math. 104 (1996), 125-133. · Zbl 0878.14018
[20] E. H. Spanier, Algebraic topology , Reprint of the 1966 original, Springer-Verlag, New-York (1989).
[21] H.-o. Tokunaga, Some examples of Zariski pairs arising from certain elliptic \(K3\) surfaces, Math. Z. 230 (2) (1999), 389-400. · Zbl 0930.14018 · doi:10.1007/PL00004697
[22] E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255-260. · Zbl 0006.41502 · doi:10.2307/2371128
[23] J-G. Yang, Sextics curves with simple singularities, Tohoku Math. J. 48 (2) (1996), 203-227. · Zbl 0866.14014 · doi:10.2748/tmj/1178225377
[24] O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929), 305-328. · JFM 55.0806.01 · doi:10.2307/2370712
[25] O. Zariski, On the irregularity of cyclic multiple planes, Ann. of Math. 32 (1931), 445-489. · Zbl 0001.40301 · doi:10.2307/1968247
[26] O. Zariski, The topological discriminant group of a Riemann surface of genus \(p\), Amer. J. Math. 59 (1937), 335-358. · Zbl 0016.32502 · doi:10.2307/2371416
[27] O. Zariski, Studies in equisingularity II. Equisingularity in codimension 1 (and characteristic zero), Amer. J. Math. 87 (1965), 972-1006. · Zbl 0146.42502 · doi:10.2307/2373257
[28] O. Zariski, Contribution to the problem of equisingularity, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Ed. Cremonese, Roma (1970), 261-343. · Zbl 0204.54503
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