Problem Session

  • Ayşe Altintaş
  • Celal Cem Sarioğlu
Conference paper
Part of the Progress in Mathematics book series (PM, volume 283)


This article contains the open problems posed at the special session of the CIMPA summer school “Arrangements, Local Systems and Singularities” held at Galatasaray University, İstanbul, 2007.


Fundamental Group Homotopy Type Hyperplane Arrangement Milnor Number Dynkin Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Abe, The stability of the family of A 2-type arrangements. J. Math. Kyoto Univ. 46 (3) (2006), 617–636.zbMATHMathSciNetGoogle Scholar
  2. [2]
    T. Abe, The stability of the family of B 2-type arrangements. Hokkaido University Preprint Series 790 (2006).Google Scholar
  3. [3]
    T. Abe and M. Yoshinaga, Splitting criterion for reflexive sheaves. Proc. of the Amer. Math. Soc. 136 (2008), 1887–1892.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    T. Abe and M. Yoshinaga, Coxeter multiarrangements with quasi-constant multiplicities. preprint arXiv:0708.3228.Google Scholar
  5. [5]
    K. Aomoto, Un théoréme du type de Matsushima-Murakami concernant l’intégrale des functions multiformes. J. Math. Pures Appl. 52 (1973), 1–11.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    V. I. Arnol’d, The cohomology ring of the colored braid group. Mat. Zametki, 5 (1960), 229–231.Google Scholar
  7. [7]
    E. Artal Bartolo and J. Carmona Ruber, Zariski pairs, fundamental groups and Alexander polynomials. J. of Math. Soc. Japan 50 (1998), 521–543.MathSciNetCrossRefGoogle Scholar
  8. [8]
    E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agustín, On sextic curves with big Milnor number. In Trends in singularities, Trends Math., 1–29, Birkhäuser, Basel, 2002.Google Scholar
  9. [9]
    E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agustín, Braid monodromy and topology of plane curves. Duke Math. J. 118 (2) (2003), 261–278.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    E. Artal Bartolo, J. Carmona Ruber and J. I. Cogolludo Agustín, Effective invariants of braid monodromy. Trans. Amer. Math. Soc. 359 (2007), 165–183.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    E. Artal Bartolo, J. Carmona Ruber, J. I. Cogolludo Agustin and M. M. Buzunáriz, Invariants of combinatorial line arrangements and Rybnikov’s example. Compositio Mathematica 141 (2005), 1578–1588.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    E. Artal Bartolo, J. I. Cogolludo Agustin and D. Matei, Arrangements of hypersurfaces and Bestvina-Brady groups. In: Mini-Workshop: Topology of closed one-forms and cohomology jumping loci, Oberwolfach Rep. 4 no. 3 (2007), 2321–2360.Google Scholar
  13. [13]
    E. Artal Bartolo, J. I. Cogolludo Agustin and H. Tokunaga, A survey on Zariski pairs. Seminario Matemático, García de galdeano, Universidad de Zaragoza, 2006.Google Scholar
  14. [14]
    W. Arvola, The fundamental group of the complement of an arrangement of complex hyperplanes. Ph.D. Thesis, University of Wisconsin-Madison, 1990.Google Scholar
  15. [15]
    C. A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes. Bull. London Math. Soc. 36 no. 3 (2004), 294–302.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    H. Bass and A. Lubotzky, Nonarithmetic superrigid groups: counterexamples to Platonov’s conjecture. Ann. of Math. 151 (2000), 1151–1173.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    A. Björner and G. Ziegler, Combinatorial stratification of complex arrangements. J. Amer. Math. Soc. 5 no. 1 (1992), 105–149.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    J. Fernandez de Bobadilla, A reformulation of Lê’s conjecture. Indag. Math. (N.S.) 17 (3) (2006), 345–352.zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    M. R. Bridson, Decision problems and profinite completions of groups. Preprint (2008).Google Scholar
  20. [20]
    M. R. Bridson, Direct factors of profinite completions and deciability. J. Group Theory (2008), to appear.Google Scholar
  21. [21]
    M. R. Bridson, The Schur multiplier, profinite completions and deciability. preprint, University of Oxford, April 2008.Google Scholar
  22. [22]
    M. R. Bridson and F. J. Grunewald, Grothendieck’s problems concerning profinite completions and representations of groups. Ann. of Math.160 (2004), 359–373.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    E. Brieskorn, Sur les groupes de tresses. In: Séminaire Bourbaki 1971/72. Lecture Notes in Math. 317, 21–44, Springer, Berlin, Heidelberg, New York, 1973.CrossRefGoogle Scholar
  24. [24]
    R.-O. Buchweitz and D. Mond, Linear free divisors and quiver representations. In: Singularities and Computer Algebra, C. Lossen and G. Pfister (Eds), London Math. Soc. Lecture Notes in Math. 324, 41–77, Cambridge University Press, 2006.Google Scholar
  25. [25]
    D. Cohen and A. Suciu, On Milnor fibrations of arrangements. J. London Math. Soc. 51 (1) (1995), 105–119.zbMATHMathSciNetGoogle Scholar
  26. [26]
    D. Cohen and A. Suciu, The braid monodromy of plane algebraic curves and hyperplane arrangements. Comment. Math. Helvetici 72 (1997), 285–315.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Cohen and A. Suciu, Characteristic varieties of arrangements. Math. Proc. Cambridge Phil. Soc. 127 (1999), 33–53.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    D. Cohen and A. Suciu, Alexander invariants of complex hyperplane arrangements. Trans. Amer. Math. Soc. 351 (1999), 4043–4067.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    D. Cohen, G. Denham and A. Suciu, Torsion in Milnor fiber homology. Algebraic and Geometric Topology 3 (2003), 511–535.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    J. Damon, Deformations of sections of singularities and Gorenstein surface singularities. Amer. J. Math. 109 (1987), 695–722.zbMATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    J. Damon, \( \mathcal{A} \)-equivalence and the equivalence of sections of images and discriminants. In: D. Mond and J. Montaldi (Eds.), Singularity Theory and Applications, Warwick 1989, Lecture Notes in Math. 1462, Springer, Berlin, Heidelberg, New York, 1991.CrossRefGoogle Scholar
  32. [32]
    J. Damon and D. Mond, \( \mathcal{A} \)-Codimension and the vanishing topology of discriminants. Invent. math. 106 (1991), 217–242.zbMATHMathSciNetCrossRefGoogle Scholar
  33. [33]
    A. Degtyarev, On deformations of singular plane sextics. J. Algebraic Geom. 17 (2008), 101–135.zbMATHMathSciNetGoogle Scholar
  34. [34]
    P. Deligne, Les immeubles des groupes de tresses génralisés. Invent. Math. 17 (1972), 273–302.zbMATHMathSciNetCrossRefGoogle Scholar
  35. [35]
    G. Denham, The Orlik-Solomon complex and Milnor fiber homology. Topology Appl. 118 (1–2) (2002), 45–63.zbMATHMathSciNetCrossRefGoogle Scholar
  36. [36]
    G. Denham, Homological aspects of hyperplane arrangements. In: F. ElZein, A. Suciu, M. Tosun, A.M. Uludağ and S. Yuzvinsky (Eds.), Lecture notes of CIMPA summer school: Arrangements, Local Systems and Singularities. Progress in Mathematics, Birkhäuser, Basel, 2010.Google Scholar
  37. [37]
    A. Dimca and A. Némethi, Hypersurface complements, Alexander modules and monodromy. Proceedings of the 7th workshop on Real and Complex Singularities, Sao Carlos, 2002; M. Ruas and T. Gaffney Eds, Contemp. Math., Amer. Math. Soc. (2004), 19–43.Google Scholar
  38. [38]
    A. Dimca and S. Yuzvinsky Lectures on Orlik-Solomon algebras. In: F. ElZein, A. Suciu, M. Tosun, A.M. Uludağ and S. Yuzvinsky (Eds.), Lecture notes of CIMPA summer school: Arrangements, Local Systems and Singularities. Progress in Mathematics, Birkhäuser, Basel, 2010.Google Scholar
  39. [39]
    I. Dolgachev and M. Kapranov, Arrangements of hyperplanes and vector bundle onn. Duke Math. J. 71 no. 3 (1993), 633–664.zbMATHMathSciNetCrossRefGoogle Scholar
  40. [40]
    N. V. Dung and H. H. Vui, The fundamental group of complex hyperplane arrangements. Acta Math. Vietnam 20 (1995), 31–41.zbMATHMathSciNetGoogle Scholar
  41. [41]
    P. H. Edelman and V. Reiner, Free arrangements and rhombic tilings. Discrete Comput. Geom. 15 no. 3 (1996), 307–340.zbMATHMathSciNetCrossRefGoogle Scholar
  42. [42]
    H. Esnault, V. Schechtman and E. Viehweg, Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109 (1992), 557–561.zbMATHMathSciNetCrossRefGoogle Scholar
  43. [43]
    E. Fadell and L. Neuwirth, Configuration spaces. Mathematica Scandinavica 10 (1962), 111–118.zbMATHMathSciNetGoogle Scholar
  44. [44]
    M. Falk, The minimal model of the complement of an arrangement of hyperplanes. Trans. Amer. Math. Soc. 309 (1988), 543–556.zbMATHMathSciNetCrossRefGoogle Scholar
  45. [45]
    M. Falk, K(π, 1) arrangements. Topology, 34 (1995), 141–154.zbMATHMathSciNetCrossRefGoogle Scholar
  46. [46]
    M. Falk, Arrangements and cohomology. Annals of Combinatorics 1 (1997), 135–157.zbMATHMathSciNetCrossRefGoogle Scholar
  47. [47]
    M. Falk, The line geometry of resonance varieties. arXiv:math/0405210v2.Google Scholar
  48. [48]
    M. Falk, Geometry and combinatorics of resonance weights. In: F. ElZein, A. Suciu, M. Tosun, A.M. Uludağ and S. Yuzvinsky (Eds.), Lecture notes of CIMPA summer school: Arrangements, Local Systems and Singularities. Progress in Mathematics, Birkhäuser, Basel, 2010.Google Scholar
  49. [49]
    M. Falk and R. Randell, The lower central series of a fiber-type arrangement. Invent. Math. 82 (1985), 77–88.zbMATHMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Falk and R. Randell, On the homotopy theory of arrangements. In: Complex Analytic Singularities. Adv. Studies in Pure Math. 8, 101–124, North Holland, 1986.Google Scholar
  51. [51]
    M. Falk and R. Randell, On the homotopy theory of arrangements II. In: M. Falk, H. Terao (Eds.) Arrangements — Tokyo 1998, Adv. Stud. Pure Math. 27, 93–25, Kinokoniyo, Tokyo, 2000.Google Scholar
  52. [52]
    M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves. Composito Math. 143 (4) (2007), 1069–1088.zbMATHMathSciNetGoogle Scholar
  53. [53]
    F.Y.C. Fung, On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory. Adv. Math. 178 (2) (2003), 244–276.zbMATHMathSciNetCrossRefGoogle Scholar
  54. [54]
    P. Gabriel, Unzerlegbare Darstellungen I. Manuscripta Math. 6 (1972), 71–103.MathSciNetCrossRefGoogle Scholar
  55. [55]
    V. Goryunov and D. Mond, Vanishing cohomology of singularities of mappings. Compositio Math. 89 (1) (1993), 45–80.zbMATHMathSciNetGoogle Scholar
  56. [56]
    M. Granger, D. Mond, A. Nieto and M. Schulze, Linear Free Divisors. submitted.Google Scholar
  57. [57]
    M. Granger and D. Mond, Linear Free Divisors and Quivers. to be published by RIMS Kôkyûroku.Google Scholar
  58. [58]
    G.-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten. Math. Ann. 250 (1980), 157–173.zbMATHMathSciNetGoogle Scholar
  59. [59]
    G.-M. Greuel, B. Martin and G. Pfister, Numerische Charakterisierung Quasihomogener Gorenstein-Kurvensingularitäten. Math. Nachr. 124 (1985), 123–131.zbMATHMathSciNetCrossRefGoogle Scholar
  60. [60]
    A. Grothendieck, Représentations linéaries et compactification profinite des proupes discrets. Manuscripta Math. 2 (1970), 375–396.zbMATHMathSciNetCrossRefGoogle Scholar
  61. [61]
    F. J. Grunewald, On some groups which can not be finitely presented. J. London Math. Soc. 17 (2) (1978), 427–436.zbMATHMathSciNetCrossRefGoogle Scholar
  62. [62]
    H. Hamm, Lokale topologische Eigenschaften komplexer Räume. Math. Ann. 191 (1971), 235–252.zbMATHMathSciNetCrossRefGoogle Scholar
  63. [63]
    A. Hattori, Topology of ℂ n minus a finite number of affine hyperplanes in general position. J. Math.Invent. math. In: Séminaire Bourbaki 1971/72. Lecture Notes in Math. 317, 21–44, Springer, Berlin, Heidelberg, New York, 1973.Google Scholar
  64. [64]
    M. Jambu and S. Papadima, A generalization of fiber type arrangements and a new deformation method. Topology 37 (1998), 1135–1164.zbMATHMathSciNetCrossRefGoogle Scholar
  65. [65]
    T. de Jong and D. van Straten, Disentanglements. In: Mond, D., Montaldi, J. (Eds) Singularity theory and its applications, Warwick 1989, Lecture Notes in Math. 1462, Springer, Berlin, 1991.CrossRefGoogle Scholar
  66. [66]
    E. R. van Kampen, On the fundamental group of an algebraic curve. Amer. J. Math. 55 (1933), 255–260.Google Scholar
  67. [67]
    T. Keilen, A claim on the rank of an injective map. Master Thesis, University of Warwick, 1993.Google Scholar
  68. [68]
    T. Keilen and D. Mond, Injective analytic maps — a counterexample to the proof. arXiv:math.AG/0409426.Google Scholar
  69. [69]
    S. Kleiman, Multiple-point formulas: I. Iteration. Acta Math. 147 (1981), 13–49.zbMATHMathSciNetCrossRefGoogle Scholar
  70. [70]
    S. Kleiman, J. Lipman and B. Ulrich, The multiple-point schemes of a finite curvilinear map of codimension one. Ark. Mat. 34 (2) (1996), 285–326.zbMATHMathSciNetCrossRefGoogle Scholar
  71. [71]
    T. Kohno, Série de Poincaré-Koszul associée aux groupes de tresses pures. Invent. Math. 82 (1985), 57–75.zbMATHMathSciNetCrossRefGoogle Scholar
  72. [72]
    Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type. Amer. J. Math. 98 (1976), 67–78.zbMATHMathSciNetCrossRefGoogle Scholar
  73. [73]
    Lê Dũng Tráng, Sur les cycles évanouissants des espaces analytiques. C. R. Acad. Sc. 288 no. 4 (1979), A283–A285.Google Scholar
  74. [74]
    A. Libgober, On the homotopy type of the complement to the plane algebraic curves. J. für Reine und Angewandte Mathematische 367 (1986), 103–114.zbMATHMathSciNetCrossRefGoogle Scholar
  75. [75]
    A. Libgober, Fundamental groups of the complements to plane singular curves. Procedings of Symposia in Pure Mathematics 46 (1987), 29–45.MathSciNetGoogle Scholar
  76. [76]
    A. Libgober, Invariants of plane algebraic curves via representations of braid groups. Inventiones Mathematicae 95 (1989), 25–30.zbMATHMathSciNetCrossRefGoogle Scholar
  77. [77]
    A. Libgober, On the homology of finite Abelian coverings. Topology and Its Applications 43 (1992), 157–166.zbMATHMathSciNetCrossRefGoogle Scholar
  78. [78]
    A. Libgober and S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems. Composito Math. 121 (2000), 337–361.zbMATHMathSciNetCrossRefGoogle Scholar
  79. [79]
    P. Lima-Filho and H. Schenck, The holonomy Lie algebras and the LCS formula for subarrangements of A n. preprint, 2008.Google Scholar
  80. [80]
    E. J. N. Looijenga, Isolated singular points on complete intersections. London Math. Soc. Lecture Notes in Math. 77, Cambridge, London: Cambridge Univ. Press, 1984.Google Scholar
  81. [81]
    E. J. N. Looijenga and J.H.M. Steenbrink, Milnor numbers and Tjurina numbers of complete intersections. Math. Ann. 271 (1985), 121–124.zbMATHMathSciNetCrossRefGoogle Scholar
  82. [82]
    P. Lorist, The geometry of B x. Indag. Math. 48 (1986), 423–442.zbMATHMathSciNetGoogle Scholar
  83. [83]
    I. Luengo and A. Pichon, Lê’s conjecture for cyclic covers. In: J.-P. Brasselet, T. Suwa (Eds.), Singularities Franco-Faponaises, Sémin. Congr., vol. 10, 163–190, Soc. Math. France, Paris, 2005.Google Scholar
  84. [84]
    W. L. Marar, The Euler characteristic of the disentanglement of the image of a corank 1 map-germ. In: D. Mond, D and J. Montaldi (Eds.), Singularity Theory and Applications, Warwick 1989, Lecture Notes in Math. 1462, Springer, Berlin, Heidelberg, New York, 1991.CrossRefGoogle Scholar
  85. [85]
    W. L. Marar, Mapping fibrations and multiple point schemes. Ph.D. Thesis, University of Warwick, 1989.Google Scholar
  86. [86]
    W. L. Marar and D. Mond, Multiple point schemes for corank 1 maps. J. London Math. Soc. 39 (2) (1989), 553–567.zbMATHMathSciNetCrossRefGoogle Scholar
  87. [87]
    J. Martinet, Singularities of Smooth Functions and Maps. Cambridge Univ. Press, 1982.Google Scholar
  88. [88]
    D. Matei and A. I. Suciu, Counting homomorphisms onto finite solvable groups, Journal of Algebra 286 no. 1 (2005), 161–186.zbMATHMathSciNetCrossRefGoogle Scholar
  89. [89]
    J. Mather, Stability of C -mappings V I: the nice dimensions. Proc. Liverpool Singularities Symposium Vol. 1, Lecture Notes in Math. 192, 207–255, Springer, Berlin, 1971.Google Scholar
  90. [90]
    J. Mather, Stability of C -mappings: IV. Classification of stable germs by Ralgebras. Inst. Hautes tudes Sci. Publ. Math. 37 (1969), 223–248.zbMATHMathSciNetCrossRefGoogle Scholar
  91. [91]
    J. Mather, Stability of C -mappings. III. Finitely determined map germs. Inst. Hautes tudes Sci. Publ. Math. 35 (1968), 279–308.MathSciNetGoogle Scholar
  92. [92]
    J. Mather, Stability of C -mappings: II. Infinitesimal stability implies stability. Ann. of Math., 89 (1969), 259–291.MathSciNetCrossRefGoogle Scholar
  93. [93]
    J. Milnor, Singular points on complex hypersurfaces. Ann. Math. Stud. vol. 61, Princeton, 1968.Google Scholar
  94. [94]
    J. Milnor and P. Orlik, Isolated singularities defined by weighted homogeneous polynomials. Topology 9 (1970), 385–393.zbMATHMathSciNetCrossRefGoogle Scholar
  95. [95]
    B. Moishezon, Stable branch curves and braid monodromies. Lecture Notes in Math. 862, 21–44, Springer, Berlin, Heidelberg, New York, 1981.Google Scholar
  96. [96]
    D. Mond, Vanishing cycles for analytic maps. In: D. Mond, D and J. Montaldi (Eds.), Lecture Notes in Math. 1462, Springer, Berlin, Heidelberg, New York, 1991.Google Scholar
  97. [97]
    D. Mond, Some remarks on the geometry and classification of germs of maps from surfaces to 3-spaces. Topology 26 (1987), 361–383.zbMATHMathSciNetCrossRefGoogle Scholar
  98. [98]
    D. Mond and R. Pellikaan, Fitting ideals and multiple points of analytic mappings. Algebraic geometry and complex analysis (Pátzcuaro, 1987), 107–161, Lecture Notes in Math. 1414, Springer, Berlin, 1989.CrossRefGoogle Scholar
  99. [99]
    M. Mustatâ and H. Schenck, The module of logarithmic p-forms of a locally free arrangement. J. Algebra 241 no. 2 (2001), 699–719.zbMATHMathSciNetCrossRefGoogle Scholar
  100. [100]
    A. Némethi, Injective analytic mappings. Duke Math. J. 69 (2) (1993), 335–347.zbMATHMathSciNetCrossRefGoogle Scholar
  101. [101]
    M. Oka, Two transforms of plane curves and their fundamental groups. J. Math. Sci. Univ. Tokyo 3 no. 2 (1996), 399–443.zbMATHMathSciNetGoogle Scholar
  102. [102]
    C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces. Progress in Mathematics 3. Birkhäuser, Boston, Mass. 1980.Google Scholar
  103. [103]
    P. Orlik, Complements of subspace arrangements. J. Alg. Geom. 1 (1992), 147–156.zbMATHMathSciNetGoogle Scholar
  104. [104]
    P. Orlik and R. Randell, The Milnor fiber of a generic arrangement. Arxiv for matematik, 31 (1993), 71–81.zbMATHMathSciNetCrossRefGoogle Scholar
  105. [105]
    P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes. Invent. Math. 56 (1980), 167–189.zbMATHMathSciNetCrossRefGoogle Scholar
  106. [106]
    P. Orlik and H. Terao, Arrangements of hyperplanes. Lecture Notes in Math. 300, Springer, Berlin, Heidelberg, New York, 1992.Google Scholar
  107. [107]
    N. G. J. Pagnon, Generic fibers of the generalized Springer resolution of type A”. Adv. Math. 194 (2) (2005), 437–462.zbMATHMathSciNetCrossRefGoogle Scholar
  108. [108]
    S. Papadima, Higher Homotopy Groups of Complements of Complex Hyperplane Arrangements. Advances in Mathematics 165 (2002), 71–100.zbMATHMathSciNetCrossRefGoogle Scholar
  109. [109]
    S. Papadima and A. Suciu, When does the associated graded Lie algebra of an arrangement group decompose?. Commentarii Mathematici Helvetici 81 (2006), 859–875.zbMATHMathSciNetCrossRefGoogle Scholar
  110. [110]
    S. Papadima and S. Yuzvinky, On rational K [π, 1] spaces and Koszul algebras. J. Pure Appl. Alg. 144 (1999), 156–167.CrossRefGoogle Scholar
  111. [111]
    L. Paris, On the fundamental group of the complement of a complex hyperplane arrangement. In: M. Falk and H. Terao (Eds.), Arrangements — Tokyo 1998, Adv. Stud. Pure Math. 27, 257–272, Kinokuniya, Tokyo, 2000.Google Scholar
  112. [112]
    V. Platonov and O. I. Tavgen, Grothendieck’s problem on profinite completions of groups. Soviet Math. Dokl. 33 (1986), 822–825.zbMATHGoogle Scholar
  113. [113]
    V. Platonov and O. I. Tavgen, Grothendieck’s problem on profinite completions and representations of groups. K theory 4 (1990), 45–47.MathSciNetGoogle Scholar
  114. [114]
    A. Postnikov and R. Stanley, Deformations of Coxeter hyperplane arrangements. J. Combin. Theory Ser. A 91 (1–2) (2000), 544–597.MathSciNetCrossRefGoogle Scholar
  115. [115]
    L. Pyber, Groups of intermediate subgroups growth and a problem of Grothendieck. Duke Math. J. 121 (2004), 169–188.zbMATHMathSciNetCrossRefGoogle Scholar
  116. [116]
    R. Randell, Milnor fibraions of lattice isotopic arrangements. Proceedings of Symposia in Pure Mathematics 40 (1983), 415–419.MathSciNetGoogle Scholar
  117. [117]
    R. Randell, The fundamental group of the complement of a union of complex hyperplanes. Invent. Math. 69 (1982), 103–108. Correction: Invent. Math. 80 (1985), 467–468.zbMATHMathSciNetCrossRefGoogle Scholar
  118. [118]
    R. Randell, Lattice-isotopic arrangements are topologicaly isomorphic. Proceedings of the American Mathematical Society 107 (1989), 555–559.zbMATHMathSciNetCrossRefGoogle Scholar
  119. [119]
    R. Randell, Homotopy and group cohomology of arrangements, Topology Appl. 78 (1997), 201–213.zbMATHMathSciNetCrossRefGoogle Scholar
  120. [120]
    C. M. Ringel, Representations of K species and bimodules. J. Alg. 41 (1976), 269–302.zbMATHMathSciNetCrossRefGoogle Scholar
  121. [121]
    G. Rybnikov, On the fundamental group of a complex hyperplane arrangement. DIMACS Technical Reports 107, 1994; arXiv:math/9805056v1.Google Scholar
  122. [122]
    K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14 (1971), 123–142.zbMATHMathSciNetCrossRefGoogle Scholar
  123. [123]
    K. Saito, Theory of logarithmic differential forms and logarithmic vector Şelds. J. Fac. Sci. Univ. Tokyo Sect. Math. 27 (1980), 265–291.zbMATHGoogle Scholar
  124. [124]
    K. Saito, On a linear structure of the quotient variety by a finite reflexion group. Publ. Res. Inst. Math. Sci. 29 (4) (1993), 535–579.zbMATHMathSciNetCrossRefGoogle Scholar
  125. [125]
    M. Salvetti, Arrangements of lines and monodromy of plane curves. In: Algebraic Geometry. Lecture Notes in Math. 862, 107–192, Springer, 1981.Google Scholar
  126. [126]
    M. Salvetti, Topology of the complement of real hyperplanes inN. Invent. Math. 88 (1987), 603–618.zbMATHMathSciNetCrossRefGoogle Scholar
  127. [127]
    H. Schenck, Elementary modifications on line configurations in2. Comment. Math. Helv. 78 (3) (2003), 447–462.zbMATHMathSciNetCrossRefGoogle Scholar
  128. [128]
    H. K. Schenck, A. I Suciu, Lower central series and free resolutions of hyperplane arrangements. Transactions of the American Mathematical Society, Vol. 354 no. 9 (2002), 3409–3433.zbMATHMathSciNetCrossRefGoogle Scholar
  129. [129]
    B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements. J. London Math. Soc. 56 (2) (1997), 477–490.zbMATHMathSciNetCrossRefGoogle Scholar
  130. [130]
    I. Shimada, A note on Zariski pairs. Composito Math. 104 (2) (1996), 125–133.zbMATHMathSciNetGoogle Scholar
  131. [131]
    I. Shimada, Fundamental groups of complements to singular plane curves. Amer. J. Math. 119 (1) (1997), 127–157.zbMATHMathSciNetCrossRefGoogle Scholar
  132. [132]
    I. Shimada, On arithmetic Zariski pairs in degree 6. Adv. Geom. 8 (2) (2008), 205–225.zbMATHMathSciNetCrossRefGoogle Scholar
  133. [133]
    I. Shimada, Non-homeomorphic conjugate complex varieties. arXiv:math/0701115v2.Google Scholar
  134. [134]
    I. Shimada, Transcendental lattices and supersingular reduction lattices of a singular K3 surface. Preprint, Trans. Amer. Math. Soc., 2008.Google Scholar
  135. [135]
    P. Slodowy, Four lectures on simple groups and singularities. Commun. Math. Inst. Rijksuniv. Utr. 11, 1980.Google Scholar
  136. [136]
    L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement. Adv. in Math. 64 (3) (1987), 305–325.zbMATHMathSciNetCrossRefGoogle Scholar
  137. [137]
    L. Solomon and H. Terao, The double Coxeter arrangement. Comm. Math. Helv. 73 (1998) 237–258.zbMATHMathSciNetCrossRefGoogle Scholar
  138. [138]
    N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold. Indag. Math. 38 (1976), 452–456.MathSciNetGoogle Scholar
  139. [139]
    T. A. Springer, The unipotent variety of a semisimple group. in: S. Abhyankar (Ed.), Proc. of the Bombay Colloqu. in Algebraic Geometry, 373–391, Oxford Univ. Press, London, 1969.Google Scholar
  140. [140]
    R. Steinberg, Conjugacy Classes in Algebraic Groups. Lecture Notes in Math. 366, Springer, Berlin, 1974.Google Scholar
  141. [141]
    R. Steinberg, An occurrence of the Robinson-Schensted correspondence. J. Algebra 113 (2) (1988), 523–528.zbMATHMathSciNetCrossRefGoogle Scholar
  142. [142]
    J. Stipins, Old and new examples of k-nets in2. arXiv:math.AG/0701046.Google Scholar
  143. [143]
    A. Suciu, Fundamental groups of line arrangements: Enumerative aspects, in: E. Priveto (ed.), Advances in algebraic geometry motivated by physics. Contemporary Math. 276, Amer. Math. Soc. (2001), 43–79.Google Scholar
  144. [144]
    H. Terao, Arrangements of hyperplanes and their freness. I. Journal of the Faculty of Science of the University of Tokyo 27 (1980), 293–320.zbMATHMathSciNetGoogle Scholar
  145. [145]
    H. Terao, Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula. Invent. Math. 63 (1) (1981), 159–179.zbMATHMathSciNetCrossRefGoogle Scholar
  146. [146]
    H. Terao, Multiderivations of Coxeter arrangements. Invent. Math. 148 (3) (2002), 659–674.zbMATHMathSciNetCrossRefGoogle Scholar
  147. [147]
    A.M. Uludağ, More Zariski pairs and finite fundamental groups of curve complements. Manuscripta Math. 106 (2001), 271–277.zbMATHMathSciNetCrossRefGoogle Scholar
  148. [148]
    G. Urzua, On line arrangements with applications to 3-nets. arXiv:0704.0469v3.Google Scholar
  149. [149]
    A. Wakamiko, On the Exponents of 2-Multiarrangements. Tokyo J. Math. 30 (2007), 99–116.zbMATHMathSciNetCrossRefGoogle Scholar
  150. [150]
    J.A. Vargas, Fixed points under the action of unipotent elements of SL n in the flag variety. Bol. Soc. Mat. Mexicana, 24 (1979), 1–14.zbMATHMathSciNetGoogle Scholar
  151. [151]
    H. Vosegaard, A characterization of quasi-homogeneous purely elliptic complete intersection singularities. Compositio Math. 124 (2000), 111–121.zbMATHMathSciNetCrossRefGoogle Scholar
  152. [152]
    H. Vosegaard, A characterization of quasi-homogeneous complete intersection singularities. J. Algebraic Geom. 11 (3) (2002), 581–597.zbMATHMathSciNetGoogle Scholar
  153. [153]
    J. Wahl, A characterization of quasi-homogeneous Gorenstein surface singularities. Compositio Math. 55 (1985), 269–288.zbMATHMathSciNetGoogle Scholar
  154. [154]
    C.T.C. Wall, Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13 (1981), 481–539.zbMATHMathSciNetCrossRefGoogle Scholar
  155. [155]
    M. Yoshinaga, The primitive derivation and freeness of multi-Coxeter arrangements. Proc. Japan Acad., 78, Ser. A (2002) 116–119.zbMATHMathSciNetCrossRefGoogle Scholar
  156. [156]
    M. Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner. Invent. Math. 157 (2) (2004), 449–454.zbMATHMathSciNetCrossRefGoogle Scholar
  157. [157]
    M. Yoshinaga, On the freeness of 3-arrangements. Bull. London Math. Soc. 37 (1) (2005), 126–134.zbMATHMathSciNetCrossRefGoogle Scholar
  158. [158]
    S. Yuzvinsky, Cohomology of Brieskorn-Orlik-Solomon algebras. Comm. Algebra 23 (1995), 5339–5354.zbMATHMathSciNetCrossRefGoogle Scholar
  159. [159]
    S. Yuzvinsky, Orlik-Solomon algebras in algebra and topology. Russian Math. Surveys 56 (2) (2001), 293–364.zbMATHMathSciNetCrossRefGoogle Scholar
  160. [160]
    S. Yuzvinsky, Realization of finite abelian groups by nets in2. Communications in Algebra 140 (6) (2004), 1614–1624.zbMATHMathSciNetGoogle Scholar
  161. [161]
    O. Zariski, On the problem of existence of algeraic functions of two variables possessing a given branch curve. Amer. J. Math. 51 (1929), 305–328.zbMATHMathSciNetCrossRefGoogle Scholar
  162. [162]
    G. Ziegler, Multiarrangements of hyperplanes and their freeness. Singularities (Iowa City, IA, 1986), 345–359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Ayşe Altintaş
    • 1
  • Celal Cem Sarioğlu
    • 2
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of MathematicsDokuz Eylül University Tinaztepe Campus Faculty of Arts and SciencesİzmirTurkey

Personalised recommendations