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Izv. Akad. Nauk SSSR Ser. Mat., 1975, Volume 39, Issue 2, Pages 278–293 (Mi izv1831)  

This article is cited in 29 scientific papers (total in 29 papers)

On Kummer surfaces

V. V. Nikulin

Abstract: In this paper we show that a Kähler $K3$ surface containing 16 nonsingular rational curves which do not intersect one another is a Kummer surface. We also give a direct proof of the global Torelli theorem for Kummer surfaces and develop a criterion for a surface to be Kummer which refines the criterion in the paper “A Torelli theorem for algebraic $K3$ surfaces” by I. I. Pyatetskii-Shapiro and I. R. Shafarevich.
Bibliography: 8 items.

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English version:
Mathematics of the USSR-Izvestiya, 1975, 9:2, 261–275

Bibliographic databases:

UDC: 513.6
MSC: Primary 14J10; Secondary 14J25
Received: 26.04.1974

Citation: V. V. Nikulin, “On Kummer surfaces”, Izv. Akad. Nauk SSSR Ser. Mat., 39:2 (1975), 278–293; Math. USSR-Izv., 9:2 (1975), 261–275

Citation in format AMSBIB
\by V.~V.~Nikulin
\paper On Kummer surfaces
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1975
\vol 39
\issue 2
\pages 278--293
\jour Math. USSR-Izv.
\yr 1975
\vol 9
\issue 2
\pages 261--275

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    This publication is cited in the following articles:
    1. Looijenga E. Peters C., “Torelli Theorems for Kahler K3 Surfaces”, Compos. Math., 42:2 (1980), 145–186  mathscinet  zmath  isi
    2. Yoichi Miyaoka, “The maximal number of quotient singularities on surfaces with given numerical invariants”, Math Ann, 268:2 (1984), 159  crossref  mathscinet  zmath  isi
    3. Eduard Looijenga, Jonathan Wahl, “Quadratic functions and smoothing surface singularities”, Topology, 25:3 (1986), 261  crossref
    4. Daniel Naie, “Surfaces d'Enriques et une construction de surfaces de type général avecp g =0”, Math Z, 215:1 (1994), 269  crossref  mathscinet  zmath  isi
    5. W. Barth, Th. Bauer, “Smooth quartic surfaces with 352 conics”, manuscripta math, 85:1 (1994), 409  crossref  mathscinet  zmath
    6. Bernd Jakob, “Poncelet 5-gons and abelian surfaces”, manuscripta math, 83:1 (1994), 183  crossref  mathscinet  zmath  isi
    7. Daniel Ruberman, “Configurations of 2-spheres in the K3 surface and other 4-manifolds”, Math Proc Camb Phil Soc, 120:2 (1996), 247  crossref  mathscinet  zmath
    8. I. A. Cheltsov, “Bounded three-dimensional Fano varieties of integer index”, Math. Notes, 66:3 (1999), 360–365  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. K. Hulek, I. Nieto, G. K. Sankaran, “Heisenberg-invariant kummer surfaces”, Proc Edin Math Soc, 43:2 (2000), 425  crossref  mathscinet  zmath
    10. J. Keum, D.-Q. Zhang, “Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds”, Journal of Pure and Applied Algebra, 170:1 (2002), 67  crossref
    11. Kharlamov V., “Overview of topological properties of real algebraic surfaces”, Algebraic Geometry and Geometric Modeling, Mathematics and Visualization, 2006, 103–117  isi
    12. Paolo Stellari, “Derived categories and Kummer varieties”, Math Z, 256:2 (2007), 425  crossref  mathscinet  zmath  isi  elib
    13. Vijay Kumar, Washington Taylor, “Freedom and constraints in the K3 landscape”, J High Energy Phys, 2009:5 (2009), 066  crossref  isi
    14. A. Kumar, “K3 Surfaces Associated with Curves of Genus Two”, Internat Math Res Notices, 2010  crossref
    15. Kristina Frantzen, “Classification of K3-surfaces with involution and maximal symplectic symmetry”, Math Ann, 2010  crossref
    16. Adrian Clingher, Charles F. Doran, “Lattice polarized K3 surfaces and Siegel modular forms”, Advances in Mathematics, 231:1 (2012), 172  crossref
    17. A. Taormina, K. Wendland, “The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24”, J. High Energ. Phys, 2013:8 (2013)  crossref
    18. V. V. Nikulin, “Kählerian K3 surfaces and Niemeier lattices. I”, Izv. Math., 77:5 (2013), 954–997  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. Lei Zhang, “Surfaces with
      $$p_g = q= 1$$
      p g = q = 1 ,
      $$K^2 = 7$$
      K 2 = 7 and non-birational bicanonical maps”, Geom Dedicata, 2014  crossref
    20. V. V. Nikulin, “Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups”, Izv. Math., 79:4 (2015), 740–794  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. Cheng M.C.N., Harrison S., “Umbral Moonshine and K3 Surfaces”, Commun. Math. Phys., 339:1 (2015), 221–261  crossref  isi
    22. V. V. Nikulin, “Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II”, Izv. Math., 80:2 (2016), 359–402  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. Garbagnati A., Sarti A., “Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action”, Rocky Mt. J. Math., 46:4 (2016), 1141–1205  crossref  mathscinet  zmath  isi  scopus
    24. V. V. Nikulin, “Classification of Picard lattices of K3 surfaces”, Izv. Math., 82:4 (2018), 752–816  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    25. Schaffler L., “K3 Surfaces With Z(2)(2) Symplectic Action”, Rocky Mt. J. Math., 48:7 (2018), 2347–2383  crossref  mathscinet  zmath  isi  scopus
    26. V. A. Krasnov, “Real Kummer surfaces”, Izv. Math., 83:1 (2019), 65–103  mathnet  crossref  crossref  adsnasa  isi  elib
    27. Roulleau X., Sarti A., “Construction of Nikulin Configurations on Some Kummer Surfaces and Applications”, Math. Ann., 373:1-2 (2019), 597–623  crossref  zmath  isi  scopus
    28. V. V. Nikulin, “Classification of degenerations and Picard lattices of Kählerian K3 surfaces with symplectic automorphism group $D_6$”, Izv. Math., 83:6 (2019), 1201–1233  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    29. Viacheslav V. Nikulin, “Classification of Degenerations and Picard Lattices of Kählerian K3 Surfaces with Symplectic Automorphism Group $C_4$”, Proc. Steklov Inst. Math., 307 (2019), 130–161  mathnet  crossref  crossref  isi  elib
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