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Surfaces of general type with geometric genus zero: a survey

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Book cover Complex and Differential Geometry

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

In the last years there have been several new constructions of surfaces of general type with p g = 0, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with p g = 0, and gives an updated list of the existing surfaces, in the case where K 2 = 1,...,7. It also focuses on certain important aspects of this classification.

Mathematics Subject Classification (2010) Primary 14J29, 14J10. Secondary 14H30, 14J80, 20F05.

The present work took place in the realm of the DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds”

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References

  1. Armstrong, M. A., On the fundamental group of an orbit space. Proc. Cambridge Philos. Soc. 61 (1965), 639–646.

    Article  MATH  MathSciNet  Google Scholar 

  2. Armstrong, M. A., The fundamental group of the orbit space of a discontinuous group. Proc. Cambridge Philos. Soc. 64 (1968), 299–301.

    Article  MATH  MathSciNet  Google Scholar 

  3. Barlow, R., Some new surfaces with p g = 0. Duke Math. J. 51 (1984), no. 4, 889–904.

    Article  MATH  MathSciNet  Google Scholar 

  4. Barlow, R., A simply connected surface of general type with p g = 0. Invent. Math. 79 (1985), no. 2, 293–301.

    Article  MATH  MathSciNet  Google Scholar 

  5. Barlow, R., Rational equivalence of zero cycles for some more surfaces with p g = 0. Invent. Math. 79 (1985), no. 2, 303–308.

    Article  MATH  MathSciNet  Google Scholar 

  6. Barth, W., Hulek, K., Peters, C., Van de Ven, A., Compact complex surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. 4. Springer-Verlag, Berlin, 2004.

    Google Scholar 

  7. Barth, W., Peters, C., Van de Ven, A., Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Springer-Verlag, Berlin, 1984.

    Google Scholar 

  8. Bauer, I., Catanese, F., Some new surfaces with p g = q = 0. The Fano Conference, 123–142, Univ. Torino, Turin, 2004.

    Google Scholar 

  9. Bauer, I., Catanese, F., A volume maximizing canonical surface in 3-space. Comment. Math. Helv. 83 (2008), no. 2, 387–406.

    Article  MATH  MathSciNet  Google Scholar 

  10. Bauer, I., Catanese, F., The moduli space of Keum-Naie surfaces. arXiv:0909.1733 (2009).

    Google Scholar 

  11. Bauer, I., Catanese, F., Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces. arXiv:0909.3699 (2009).

    Google Scholar 

  12. Bauer, I., Catanese, F., Burniat surfaces. II. Secondary Burniat surfaces form three connected components of the moduli space. Invent. Math. 180 (2010), no. 3, 559–588.

    Article  MATH  MathSciNet  Google Scholar 

  13. Bauer, I., Catanese, F., Chan, M., Inoue surfaces with K 2 = 7. In preparation.

    Google Scholar 

  14. Bauer, I., Catanese, F., Grunewald, F., Beauville surfaces without real structures. Geometric methods in algebra and number theory, 1–42, Progr. Math., 235, Birkhäuser Boston, Boston, MA, 2005.

    Google Scholar 

  15. Bauer, I., Catanese, F., Grunewald, F., The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type. arXiv:0706.1466 (2007)

    Google Scholar 

  16. Bauer, I., Catanese, F., Grunewald, F., The classification of surfaces with p g = q = 0 isogenous to a product of curves. Pure Appl. Math. Q. 4 (2008), no. 2, part 1, 547–586

    MATH  MathSciNet  Google Scholar 

  17. Bauer, I., Catanese, F., Grunewald, F., Pignatelli, R., Quotients of products of curves, new surfaces with p g = 0 and their fundamental groups. arXiv:0809.3420 (2008), to appear in Amer. J. Math.

    Google Scholar 

  18. Bauer, I., Catanese, F., Pignatelli, R., Complex surfaces of general type: some recent progress. Global aspects of complex geometry, 1–58, Springer, Berlin, 2006.

    Google Scholar 

  19. Bauer, I., Pignatelli, R., The classification of minimal product-quotient surfaces with p g = 0. arXiv:1006.3209 (2010)

    Google Scholar 

  20. Beauville, A., Surfaces algébriques complexes. Asterisque 54, Société Mathématique de France, Paris, 1978. iii+172 pp.

    Google Scholar 

  21. Beauville, A., A Calabi-Yau threefold with non-abelian fundamental group. New trends in algebraic geometry (Warwick, 1996), 13–17, London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  22. Bloch, S., K 2 of Artinian Q-algebras, with application to algebraic cycles. Comm. Algebra 3 (1975), 405–428.

    Article  MATH  MathSciNet  Google Scholar 

  23. Bloch, S., Kas, A., Lieberman, D., Zero cycles on surfaces with p g = 0. Compositio Math. 33 (1976), no. 2, 135–145.

    MATH  MathSciNet  Google Scholar 

  24. Bombieri, E., Canonical models of surfaces of general type. Inst. Hautes Études Sci. Publ. Math. No. 42 (1973), 171–219.

    Article  MathSciNet  Google Scholar 

  25. Bombieri, E., Catanese, F., The tricanonical map of a surface with K 2 = 2, p g = 0. C. P. Ramanujam—a tribute, pp. 279–290, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978.

    Google Scholar 

  26. Bosma, W., Cannon, J., Playoust, C., The Magma algebra system. I. The user language. J. Symbolic Comput., 24 (3–4):235–265, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  27. Breuer, T., Characters and Automorphism Groups of Compact Riemann Surfaces. London Math. Soc. Lecture Note Series 280, Cambridge University Press, Cambridge, 2000. xii+199 pp.

    Google Scholar 

  28. Burniat, P., Sur les surfaces de genre P 12 > 1. Ann. Mat. Pura Appl. (4) 71 (1966), 1–24.

    MATH  MathSciNet  Google Scholar 

  29. Burns, D. M., Jr., Wahl, Jonathan M., Local contributions to global deformations of surfaces. Invent. Math. 26 (1974), 67–88.

    Article  MATH  MathSciNet  Google Scholar 

  30. Campedelli, L., Sopra alcuni piani doppi notevoli con curve di diramazione del decimo ordine. Atti Acad. Naz. Lincei 15 (1932), 536–542.

    MATH  Google Scholar 

  31. Cartwright, D. I., Steger, T., Enumeration of the 50 fake projective planes. C. R. Math. Acad. Sci. Paris 348 (2010), no. 1–2, 11–13.

    MATH  MathSciNet  Google Scholar 

  32. Castelnuovo, G., Sulle superficie di genere zero. Memorie della Soc.It. delle Scienze (detta dei XL), ser. III, t. 10, (1896).

    Google Scholar 

  33. Catanese, F., Pluricanonical mappings of surfaces with K 2 = 1, 2, q = p g = 0. in ‘C.I.M.E. 1977: Algebraic surfaces’, Liguori, Napoli (1981), 247–266.

    Google Scholar 

  34. Catanese, F., Babbage’s conjecture, contact of surfaces, symmetric determinantal varieties and applications. Invent. Math.63 (1981), no. 3, 433–465.

    Article  MATH  MathSciNet  Google Scholar 

  35. Catanese, F., On the moduli spaces of surfaces of general type. J. Differential Geom. 19 (1984), no. 2, 483–515.

    MATH  MathSciNet  Google Scholar 

  36. Catanese, F., Everywhere nonreduced moduli spaces. Invent. Math. 98 (1989), no. 2, 293–310.

    Article  MATH  MathSciNet  Google Scholar 

  37. Catanese, F., Singular bidouble covers and the construction of interesting algebraic surfaces. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 97–120, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  38. Catanese, F., Fibred Surfaces, varieties isogeneous to a product and related moduli spaces. Amer. J. Math. 122 (2000), no. 1, 1–44.

    MATH  MathSciNet  Google Scholar 

  39. Catanese, F., Moduli spaces of surfaces and real structures. Ann. of Math. (2) 158 (2003), no. 2, 577–592.

    Article  MATH  MathSciNet  Google Scholar 

  40. Catanese, F., Fibred Kähler and quasi-projective groups. Special issue dedicated to Adriano Barlotti. Adv. Geom. 2003, suppl., S13–S27.

    Google Scholar 

  41. Catanese, F., Differentiable and deformation type of algebraic surfaces, real and symplectic structures. Symplectic 4-manifolds and algebraic surfaces, 55–167, Lecture Notes in Math. 1938, Springer, Berlin, 2008.

    Google Scholar 

  42. Catanese, F., Ciliberto, C., Surfaces with p g = q = 1. Problems in the theory of surfaces and their classification (Cortona, 1988), 49–79, Sympos. Math., XXXII, Academic Press, London, 1991.

    Google Scholar 

  43. Catanese, F., Franciosi, M., Hulek, K., Reid, M., Embeddings of curves and surfaces, Nagoya Math. J. 154(1999), 185–220.

    MATH  MathSciNet  Google Scholar 

  44. Catanese, F., Kollár, J., Trento examples 2. Classification of irregular varieties, 136–139, Lecture Notes in Mathematics, 1515, Springer, Berlin, 1992.

    Google Scholar 

  45. Catanese, F., LeBrun, C., On the scalar curvature of Einstein manifolds. Math. Res. Lett. 4 (1997), no. 6, 843–854.

    MATH  MathSciNet  Google Scholar 

  46. Craighero, P.C., Gattazzo, R.Quintic surfaces of P 3 having a nonsingularmodel with q = p g = 0, P 2 ≠ 0. Rend. Sem. Mat. Univ. Padova 91 (1994), 187–198.

    MathSciNet  Google Scholar 

  47. Dedieu, T., Perroni, F., The fundamental group of quotients of a product of curves. arXiv:1003.1922 (2010)

    Google Scholar 

  48. Dolgachev, I., Algebraic surfaces with q = p g = 0. in ‘C.I.M.E. 1977: Algebraic surfaces’, Liguori, Napoli (1981), 97–215.

    Google Scholar 

  49. Dolgachev, I., Werner, C., A simply connected numerical Godeaux surface with ample canonical class. J. Algebraic Geom. 8 (1999), no. 4, 737–764. Erratum ibid, 10 (2001), no. 2, 397.

    MATH  MathSciNet  Google Scholar 

  50. Enriques, F., Introduzione alla geometria sopra le superficie algebriche. Memorie della Societa’ Italiana delle Scienze (detta “dei XL”), s.3, to. X, (1896), 1–81.

    Google Scholar 

  51. Enriques, F., Memorie scelte di geometria, vol. I, II, III. Zanichelli, Bologna, 1956, 541 pp., 1959, 527 pp., 1966,456 pp.

    Google Scholar 

  52. Francia, P., On the base points of the bicanonical system. Problems in the theory of surfaces and their classification (Cortona, 1988), 141–150, Sympos. Math., XXXII, Academic Press, London,1991

    Google Scholar 

  53. Frapporti, D., private communication (2010).

    Google Scholar 

  54. Freedman, M., The topology of four-dimensional manifolds. J. Differential Geom. 17 (1982), no. 3, 357–453.

    MATH  MathSciNet  Google Scholar 

  55. Friedman, R., Qin, Z., The smooth invariance of the Kodaira dimension of a complex surface. Math. Res. Lett. 1 (1994), no. 3, 369–376.

    MATH  MathSciNet  Google Scholar 

  56. Fujiki, A., On resolutions of cyclic quotient singularities. Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 1, 293–328.

    Article  MathSciNet  Google Scholar 

  57. Gieseker, D., Global moduli for surfaces of general type. Invent. Math. 43 (1977), no. 3, 233–282.

    Article  MATH  MathSciNet  Google Scholar 

  58. Godeaux, L., Les surfaces algébriques non rationnelles de genres arithmétique et geométrique nuls. Hermann & Cie., Paris, 1934. 33 pp.

    MATH  Google Scholar 

  59. Godeaux, L. Les involutions cycliques appartenant à une surface algébrique. Actual. Sci. Ind., 270, Hermann, Paris, 1935.

    Google Scholar 

  60. Grauert, H., Remmert, R., Komplexe Räume. Math. Ann. 136 (1958), 245–318.

    Article  MATH  MathSciNet  Google Scholar 

  61. Grunewald, F., Jaikin-Zapirain, A., Zalesskii, P. A., Cohomological goodness and the profinite completion of Bianchi groups. Duke Math. J. 144 (2008), no. 1, 53–72.

    Article  MATH  MathSciNet  Google Scholar 

  62. Guletskii, V., Pedrini, C., Finite-dimensional motives and the conjectures of Beilinson and Murre. Special issue in honor of Hyman Bass on his seventieth birthday. Part III. K-Theory 30 (2003), no. 3, 243–263.

    Google Scholar 

  63. Horikawa, E., On deformations of quintic surfaces. Invent. Math. 31 (1975), no. 1, 43–85.

    Article  MATH  MathSciNet  Google Scholar 

  64. Huppert, B., Endliche Gruppen.I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin-New York, 1967. xii+793 pp.

    Google Scholar 

  65. Inose, H., Mizukami, M., Rational equivalence of 0-cycles on some surfaces of general type with p g = 0. Math. Ann. 244 (1979), no. 3, 205–217.

    Article  MATH  MathSciNet  Google Scholar 

  66. Inoue, M., Some new surfaces of general type. Tokyo J. Math. 17 (1994), no. 2, 295–319.

    Article  MATH  MathSciNet  Google Scholar 

  67. Ishida, M.-N., Kato, F., The strong rigidity theorem for non-Archimedean uniformization. Tohoku Math. J. (2) 50 (1998), no. 4, 537–555.

    Article  MATH  MathSciNet  Google Scholar 

  68. Jost, J., Yau, S.-T., A strong rigidity theorem for a certain class of compact complex analytic surfaces. Math. Ann. 271 (1985), no. 1, 143–152.

    Article  MATH  MathSciNet  Google Scholar 

  69. Keum, J., Some new surfaces of general type with p g = 0. Unpublished manuscript, 1988.

    Google Scholar 

  70. Keum, J., A fake projective plane with an order 7 automorphism. Topology 45 (2006), no. 5, 919–927.

    Article  MATH  MathSciNet  Google Scholar 

  71. Keum, J., Quotients of fake projective planes. Geom. Topol. 12 (2008), no. 4, 2497–2515.

    Article  MATH  MathSciNet  Google Scholar 

  72. Keum, J., Lee, Y., Construction of surfaces of general type from elliptic surfaces via ℚ-Gorenstein smoothing. arXiv:1008.1222 (2010)

    Google Scholar 

  73. Kimura, S., Chow groups are finite dimensional, in some sense. Math. Ann. 331 (2005), no. 1, 173–201.

    Article  MATH  MathSciNet  Google Scholar 

  74. Klingler, B., Sur la rigidité de certains groupes fondamentaux, ľarithméticité des réseaux hyperboliques complexes, et les “faux plans projectifs”. Invent. Math. 153 (2003), no. 1, 105–143.

    Article  MATH  MathSciNet  Google Scholar 

  75. Kuga, M., FAFA Note. (1975).

    Google Scholar 

  76. Kulikov, V., Old examples and a new example of surfaces of general type with p g = 0. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 5, 123–170; translation in Izv. Math. 68 (2004), no. 5, 965–1008

    Google Scholar 

  77. Kulikov, V. S., Kharlamov, V. M., On real structures on rigid surfaces. Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 1, 133–152; translation in Izv. Math. 66 (2002), no. 1, 133–150.

    MATH  MathSciNet  Google Scholar 

  78. Lee, Y., Park, J., A simply connected surface of general type with p g = 0 and K 2 = 2. Invent. Math. 170 (2007), no. 3, 483–505.

    Article  MATH  MathSciNet  Google Scholar 

  79. Lee, Y., Park, J., A complex surface of general type with p g = 0, K 2 = 2 and H 1 = ℤ/2ℤ. Math. Res. Lett. 16 (2009), no. 2, 323–330.

    MATH  MathSciNet  Google Scholar 

  80. Manetti, M., On the moduli space of diffeomorphic algebraic surfaces. Invent. Math. 143 (2001), no. 1, 29–76

    Article  MATH  MathSciNet  Google Scholar 

  81. Manetti, M., Surfaces of Albanese general type and the Severi conjecture. Math. Nachr. 261/262 (2003), 105–122.

    Article  MathSciNet  Google Scholar 

  82. Mendes Lopes, M., Pardini, R., The bicanonical map of surfaces with p g = 0 and K 2 ≥ 7. Bull. London Math. Soc. 33 (2001), no. 3, 265–274.

    Article  MATH  MathSciNet  Google Scholar 

  83. Mendes Lopes, M., Pardini, R., A connected component of the moduli space of surfaces with p g = 0. Topology 40 (2001), no. 5, 977–991.

    Article  MATH  MathSciNet  Google Scholar 

  84. Mendes Lopes, M., Pardini, R., A survey on the bicanonical map of surfaces with p g = 0 and K 2 ≥ 2. Algebraic geometry, 277–287, de Gruyter, Berlin 2002.

    Google Scholar 

  85. Mendes Lopes, M., Pardini, R., A new family of surfaces with p g = 0 and K 2 = 3. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 4, 507–531.

    MATH  MathSciNet  Google Scholar 

  86. Mendes Lopes, M., Pardini, R., Surfaces of general type with p g = 0, K 2 = 6 and non birational bicanonical map. Math. Ann. 329 (2004), no. 3, 535–552.

    Article  MATH  MathSciNet  Google Scholar 

  87. Mendes Lopes, M., Pardini, R.. The degree of the bicanonical map of a surface with p g = 0. Proc. Amer. Math. Soc. 135 (2007), no. 5, 1279–1282.

    Article  MATH  MathSciNet  Google Scholar 

  88. Mendes Lopes, M., Pardini, R., On the algebraic fundamental group of surfaces with K 2 ≤ 3χ. J. Differential Geom. 77 (2007), no. 2, 189–199.

    MATH  MathSciNet  Google Scholar 

  89. Mendes Lopes, M., Pardini, R., Numerical Campedelli surfaces with fundamental group of order 9. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 457–476.

    Article  MATH  MathSciNet  Google Scholar 

  90. Mendes Lopes, M., Pardini, R., Reid, M., Campedelli surfaces with fundamental group of order 8. Geom. Dedicata 139 (2009), 49–55.

    Article  MATH  MathSciNet  Google Scholar 

  91. Miyaoka, Y. Tricanonical maps of numerical Godeaux surfaces. Invent. Math. 34 (1976), no. 2, 99–111.

    Article  MATH  MathSciNet  Google Scholar 

  92. Miyaoka, Y., On numerical Campedelli surfaces. Complex analysis and algebraic geometry, 113–118. Iwanami Shoten, Tokyo, 1977.

    Google Scholar 

  93. Miyaoka, Y. On the Chern numbers of surfaces of general type. Invent. Math. 42 (1977), 225–237.

    Article  MathSciNet  Google Scholar 

  94. Miyaoka, Y. Algebraic surfaces with positive indices. Classification of algebraic and analytic manifolds (Katata, 1982), 281–301, Progr. Math. 39, Birkhäuser Boston, Boston, MA, 1983.

    Google Scholar 

  95. Mostow, G. D. Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, No. 78. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973.

    Google Scholar 

  96. Mumford, D., Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ. 9 (1968), 195–204.

    MathSciNet  Google Scholar 

  97. Mumford, D., An algebraic surface with K ample, (K 2) = 9, p g = q = 0. Amer. J. Math. 101 (1979), no. 1, 233–244.

    Article  MATH  MathSciNet  Google Scholar 

  98. Naie, D., Surfaces d‘Enriques et une construction de surfaces de type général avec p g = 0. Math. Z. 215 (1994), no. 2, 269–280.

    Article  MATH  MathSciNet  Google Scholar 

  99. Naie, D., Numerical Campedelli surfaces cannot have the symmetric group as the algebraic fundamental group. J. London Math. Soc. (2) 59 (1999), no. 3, 813–827.

    Article  MathSciNet  Google Scholar 

  100. Neves, J., Papadakis, S.A., A construction of numerical Campedelli surfaces with torsion ℤ/6. Trans. Amer. Math. Soc. 361 (2009), no. 9, 4999–5021.

    Article  MATH  MathSciNet  Google Scholar 

  101. Oort, F., Peters, C., A Campedelli surface with torsion group Z/2. Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 4, 399–407.

    MATH  MathSciNet  Google Scholar 

  102. Pardini, R., Abelian covers of algebraic varieties. J. Reine Angew. Math. 417 (1991), 191–213.

    Article  MATH  MathSciNet  Google Scholar 

  103. Pardini, R., The Severi inequality K 2 ≥ 4χ for surfaces of maximal Albanese dimension. Invent. Math. 159 (2005), no. 3, 669–672.

    Article  MATH  MathSciNet  Google Scholar 

  104. Park, H., A complex surface of general type with p g = 0, K 2 = 4, and π 1 = ℤ/2ℤ. arXiv:0910.3476 (2009)

    Google Scholar 

  105. Park, H., Park, J., Shin, D., A complex surface of general type with p g = 0, K 2 = 3 and H 1 = ℤ/2ℤ. arXiv:0803.1322 (2008)

    Google Scholar 

  106. Park, H., Park, J., Shin, D., A simply connected surface ofgeneral type with p g = 0 and K 2 = 3. Geom. Topol. 13 (2009), no. 2, 743–767.

    Article  MATH  MathSciNet  Google Scholar 

  107. Park, H., Park, J., Shin, D., A simply connected surface of general type with p g = 0 and K 2 = 4. Geom. Topol. 13 (2009), no. 3, 1483–1494.

    Article  MATH  MathSciNet  Google Scholar 

  108. Penegini, M., The classification of isotrivial fibred surfaces with p g = q = 2. arXiv:0904.1352 (2009)

    Google Scholar 

  109. Peters, C. A. M., On two types of surfaces of general type with vanishing geometric genus. Invent. Math. 32 (1976), no. 1, 33–47.

    Article  MATH  MathSciNet  Google Scholar 

  110. Peters, C. A. M., On certain examples of surfaces with p g = 0 due to Burniat. Nagoya Math. J. 66 (1977), 109–119.

    MathSciNet  Google Scholar 

  111. Prasad, G., Yeung, S., Fake projective planes Invent. Math. 168 (2007), no. 2, 321–370.

    Article  MATH  MathSciNet  Google Scholar 

  112. Prasad, G., Yeung, S., Addendum to “Fake projective planes”. arXiv:0906.4932 (2009)

    Google Scholar 

  113. Polizzi, F., Standard isotrivial fibrations with p g = q = 1. J. Algebra 321 (2009), no. 6, 1600–1631.

    Article  MATH  MathSciNet  Google Scholar 

  114. Polizzi, F., Numerical properties of isotrivial fibrations. Geom. Dedicata 147 (2010), no. 1, 323–355.

    Article  MATH  MathSciNet  Google Scholar 

  115. Reid, M., Surfaces with p g = 0, K 2 = 2. Preprint available at http://www.warwick.ac.uk/ masda/surf/

  116. Reid, M., Surfaces with p g = 0, K 2 = 1. J. Fac. Sci. Tokyo Univ. 25 (1978), 75–92.

    MATH  Google Scholar 

  117. Reid, M., π 1 for surfaces with small K 2. Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), 534–544, Lecture Notes in Math., 732, Springer, Berlin, 1979.

    Google Scholar 

  118. Reid, M., Young person’s guide to canonical singularities. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345–414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.

    Google Scholar 

  119. Reid, M., Campedelli versus Godeaux. Problems in the theory of surfaces and their classification (Cortona, 1988), 309–365, Sympos. Math., XXXII, Academic Press, London, 1991

    Google Scholar 

  120. Reider, I., Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. of Math. (2) 127 (1988), no. 2, 309–316.

    Article  MathSciNet  MATH  Google Scholar 

  121. Segal, D., Decidable properties ofpolycyclic groups. Proc. London Math. Soc. (3) 61 (1990), no. 3, 497–528.

    Article  MATH  MathSciNet  Google Scholar 

  122. Serrano, F., Isotrivial fibred surfaces. Ann. Mat. Pura Appl. (4) 171 (1996), 63–81.

    Article  MATH  MathSciNet  Google Scholar 

  123. Serre, J.P., Exemples de variétés projectives conjuguées non homéomorphes. C. R. Acad. Sci. Paris 258 (1964), 4194–4196.

    MATH  MathSciNet  Google Scholar 

  124. Serre, J. P., Cohomologie galoisienne. Fifth edition. Lecture Notes in Mathematics, 5. Springer-Verlag, Berlin, 1994. x+181 pp.

    Google Scholar 

  125. Shavel, I. H., A class of algebraic surfaces of general type constructed from quaternion algebras. Pacific J. Math. 76 (1978), no. 1, 221–245.

    MATH  MathSciNet  Google Scholar 

  126. Supino, P., A note on Campedelli surfaces. Geom. Dedicata 71 (1998), no. 1, 19–31.

    Article  MATH  MathSciNet  Google Scholar 

  127. Toledo, D., Projective varieties with non-residually finite fundamental group. Inst. Hautes Études Sci. Publ. Math. No. 77 (1993), 103–119.

    Article  MATH  MathSciNet  Google Scholar 

  128. Vakil, R., Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164 (2006), no. 3, 569–590.

    Article  MATH  MathSciNet  Google Scholar 

  129. Voisin, C., Sur les zéro-cycles de certaines hypersurfaces munies d’un automor- phisme. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4, 473–492.

    MATH  MathSciNet  Google Scholar 

  130. Werner, C., A surface of general type with p g = q = 0, K 2 = 1. Manuscripta Math. 84 (1994), no. 3–4, 327–341.

    Article  MATH  MathSciNet  Google Scholar 

  131. Werner, C., A four-dimensional deformation of a numerical Godeaux surface. Trans. Amer. Math. Soc. 349 (1997), no. 4, 1515–1525.

    Article  MATH  MathSciNet  Google Scholar 

  132. Xiao, G., Surfaces fibrées en courbes de genre deux. Lecture Notes in Mathematics, 1137. Springer-Verlag, Berlin, 1985.

    Google Scholar 

  133. Xiao, G., Finitude de l’application bicanonique des surfaces de type général. Bull. Soc. Math. France 113 (1985), no. 1, 23–51.

    MATH  MathSciNet  Google Scholar 

  134. Yau, S.T., Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. U.S.A. 74 (1977) no. 5, 1798–1799.

    Article  Google Scholar 

  135. Yau, S.T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl Mathematik VIII, Mathematisches Institut der, Universitäat Bayreuth, Universitäatsstr. 30, D-95447, Bayreuth, Germany

    Ingrid Bauer & Fabrizio Catanese

  2. Dipartimento di Matematica della, Università di Trento, Via Sommarive 14, I-38123, Trento, TN, Italy

    Roberto Pignatelli

Authors
  1. Ingrid Bauer
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  2. Fabrizio Catanese
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  3. Roberto Pignatelli
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Correspondence to Ingrid Bauer .

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Editors and Affiliations

  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

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Bauer, I., Catanese, F., Pignatelli, R. (2011). Surfaces of general type with geometric genus zero: a survey. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_1

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