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Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

In beloved memory of Paolo (De Bartolomeis)
  • Ingrid Bauer
  • Fabrizio Catanese
Article
  • 3 Downloads

Abstract

We prove the equisingular rigidity of the singular Hirzebruch–Kummer coverings X(n, \(\mathcal {L}\)) of the projective plane branched on line configurations \(\mathcal {L}\), satisfying some technical condition. In the case, \(\mathcal {L}=\) the complete quadrangle, we give explicit equations of the Hirzebruch–Kummer covering \(S_n(=\) the minimal desingularisation of \(X (n, \mathcal {L}))\) in a product of four Fermat curves of degree n. Since \(S_n\) is the \((\mathbb {Z}/n)^5\) covering of the Del Pezzo surface \(Y_5\) of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of \(Y_5\) in \((\mathbb {P}^1)^4\).We describe more generally determinantal equations for all Del Pezzo surfaces of degree \(9-k \le 6\) as subvarieties of the k-fold product of the projective line.

Keywords

Rigid complex manifolds and varieties Branched coverings Hirzebruch Kummer coverings Deformation theory Configurations of lines Del Pezzo surfaces 

Mathematics Subject Classification

14B12 14J15 14J29 14J45 14E20 14F17 32G05 32S15 32J15 52C35 

References

  1. 1.
    Barthel, G., Hirzebruch, F., Höfer, T.: Geradenkonfigurationen und algebraische Flächen. Aspects of Mathematics, D4. Friedr. Vieweg & Sohn, Braunschweig (1987). (German) CrossRefGoogle Scholar
  2. 2.
    Bauer, I.C., Catanese, F.: A volume maximizing canonical surface in 3-space. Comment. Math. Helv. 83(2), 387–406 (2008).  https://doi.org/10.4171/CMH/129 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bauer, I., Catanese, F.: Burniat surfaces III: deformations of automorphisms and extended Burniat surfaces. Doc. Math. 18, 1089–1136 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bauer, I., Catanese, F.: On rigid compact complex surfaces and manifolds. Adv. Math. 333, 620–629 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Buchsbaum, D.A., Eisenbud, D.: What makes a complex exact? J. Algebra 25, 259–268 (1973)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Catanese, F.: Moduli of Algebraic Surfaces, Theory of Moduli (Montecatini Terme, 1985). Lecture Notes in Mathematics, vol. 1337, pp. 1–83. Springer, Berlin (1988).  https://doi.org/10.1007/BFb0082806 CrossRefGoogle Scholar
  7. 7.
    Catanese, F.: Kodaira fibrations and beyond: methods for moduli theory. Jpn. J. Math. 12(2), 91–174 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Catanese, F., Dettweiler, M.: The direct image of the relative dualizing sheaf needs not be semiample. C. R. Math. Acad. Sci. Paris 352(3), 241–244 (2014).  https://doi.org/10.1016/j.crma.2013.12.015 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Catanese, F., Dettweiler, M.: Answer to a question by Fujita on Variation of Hodge Structures. Adv. Stud. Pure. Math. 74, 73–102 (2017). (Higher Dimensional Algebraic Geometry - in honour of Professor Yujiro Kawamata’s sixtieth birthday)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Catanese, F., Dettweiler, M.: Vector bundles on curves coming from variation of Hodge structures. Int. J. Math. 27(7), 1640001, 25 (2016).  https://doi.org/10.1142/S0129167X16400012 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kodaira, K.: On stability of compact submanifolds of complex manifolds. Am. J. Math. 85, 79–94 (1963)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mostow, G.D., Siu, Y.T.: A compact Kähler surface of negative curvature not covered by the ball. Ann. Math. (2) 112(2), 321–360 (1980).  https://doi.org/10.2307/1971149 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Panov, D.: Complex surfaces with CAT(0) metrics. Geom. Funct. Anal. 21(5), 1218–1238 (2011).  https://doi.org/10.1007/s00039-011-0133-8 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sernesi, E.: Small deformations of global complete intersections. Boll. Un. Mat. Ital. (4) 12(1–2), 138–146 (1975). (English, with Italian summary) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zheng, F.: Hirzebruch–Kato surfaces, Deligne–Mostow’s construction, and new examples of negatively curved compact Kähler surfaces. Commun. Anal. Geom. 7(4), 755–786 (1999).  https://doi.org/10.4310/CAG.1999.v7.n4.a4 CrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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