Rocky Mountain Journal of Mathematics

Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action

Alice Garbagnati and Alessandra Sarti

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In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number~17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces, and we use them to describe projective models of Kummer surfaces of $(1,d)$-polarized abelian surfaces for $d=1,2,3$. As a consequence, we prove that, in these cases, the N\'eron-Severi group can be generated by lines.

In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group $(\mathbb{Z} /2\mathbb{Z} )^4$. In particular, we describe the possible N\'eron-Severi groups of the latter in the case that the Picard number is $16$, which is the minimal possible. We also describe the N\'eron-Severi groups of the minimal resolution of the quotient surfaces which have 15 nodes. We extend certain classical results on Kummer surfaces to these families.

Article information

Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1141-1205.

First available in Project Euclid: 19 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14J10: Families, moduli, classification: algebraic theory 14J50: Automorphisms of surfaces and higher-dimensional varieties

Kummer surfaces K3 surfaces automorphisms Enriques involutions even sets of nodes


Garbagnati, Alice; Sarti, Alessandra. Kummer surfaces and K3 surfaces with $(\mathbb{Z} /2\mathbb{Z} )^4$ symplectic action. Rocky Mountain J. Math. 46 (2016), no. 4, 1141--1205. doi:10.1216/RMJ-2016-46-4-1141.

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  • V.V. Alexeev and V.V. Nikulin, Del Pezzo and K3 surfaces, Math. Soc. Japan Memoirs 15, Tokyo, 2006.
  • W. Barth, Abelian surface with a $(1,2)$-Polarization, Adv. Stud. Pure Math. 10, (1987), 41–84.
  • ––––, Even sets of eight rational curves on a K3 surface, in Complex geometry, Springer, Berlin, 2002.
  • W. Barth, K. Hulek, C. Peters and A. van de Ven, Compact complex surfaces, Second edition, Ergeb. Math. Grenzg. Folge, Springer, Berlin, 2004.
  • G. Bini and A. Garbagnati, Quotients of the Dwork pencil, J. Geom. Phys. 75 (2014), 173–198.
  • Ch. Birkenhake and H. Lange, Complex abelian varieties, Second edition, Grundl. Math. Wissen. 302, Springer-Verlag, Berlin, 2004.
  • S. Boissière and A. Sarti, On the Neron-Severi group of surfaces with many lines, Proc. Amer. Math. Soc. 136 (2008), 3861–3867.
  • D. Eklund, Curves of Heisenberg invariant quartic surfaces in projective $3$-spaces, arXiv:1010.4058v2.
  • F. Galluzzi and G. Lombardo, Correspondences between K3 surfaces, Michigan Math. J. 52 (2004), 267–277.
  • A. Garbagnati, Symplectic automorphisms on K3 surfaces, Ph.D. thesis, University of Milan, Italy, 2008, available at
  • ––––, Symplectic automorphisms on Kummer surfaces, Geom. Ded. 145 (2010), 219–232.
  • ––––, Elliptic K3 surfaces with abelian and dihedral groups of symplectic automorphisms, Comm. Alg. 41 (2013), 583–616.
  • A. Garbagnati and A. Sarti, Projective models of K3 surfaces with an even set, Adv. Geom. 8 (2008), 413–440.
  • A. Garbagnati and A. Sarti, Elliptic fibrations and symplectic automorphisms on K3 surfaces, Comm. Alg. 37 (2009), 3601–3631.
  • B. van Geemen and A. Sarti, Nikulin involutions on K3 surfaces, Math. Z. 255 (2007), 731–753.
  • R.L. Griess, Jr. and C.H. Lam, $EE_8$-lattices and dihedral groups, Pure Appl. Math. Quart. 7 (2011), 621–743.
  • P. Griffiths and J. Harris, Principles of algebraic geometry, Pure Appl. Math., Wiley-Interscience, John Wiley & Sons, New York, 1978.
  • K. Hashimoto, Finite symplectic actions on the K3 lattice, Nagoya Math. J. 206 (2012), 99–153.
  • E. Horikawa, On the periods of Enriques surfaces, I and II, Math. Annal. 234 (1978), 73–88; 235 (1978), 217–246.
  • R.W.H.T. Hudson, Kummer's quartic surfaces, Cambridge University Press, Cambridge, 1990 (revised reprint of the 1905 original).
  • K. Hulek and M. Schütt, Enriques surfaces and Jacobian elliptic K3 surfaces, Math. Z. 268 (2011), 1025–1056.
  • B. Hunt, The geometry of some special arithmetic quotients, Lect. Notes Math. 1637, Springer-Verlag, Berlin, 1996.
  • H. Inose, On certain Kummer surfaces which can be realized as non-singular quartic surfaces in $\PP^3$, J. Fac. Sci. Univ. Tokyo 23 (1976), 545–560.
  • J. Keum, Automorphisms of Jacobian Kummer surfaces, Compos. Math. 107 (1997), 269–288.
  • ––––, Every algebraic Kummer surfaces is the K3-cover of an Enriques surface, Nagoya Math. J. 118 (1990), 99–110.
  • J. Keum and S. Kondo The automorphism groups of Kummer surfaces associated to the product of two elliptic curves, Trans. Amer. Math. Soc. 353 (2001), 1469–1487.
  • K. Koike, Elliptic K3 surfaces admitting a Shioda-Inose structure, Comm. Math., Univ. St. Paul. 61 (2012), 77–86.
  • K. Koike, H. Shiga, N. Takayama and T. Tsutsui, Study on the family of K3 surfaces induced from the lattice $(D\sb 4)^3\oplus\langle-2\rangle\oplus\langle 2\rangle$, Int. J. Math. 12 (2001), 1049–1085.
  • S. Kondo, The automorphism group of a generic Jacobian Kummer surface, J. Alg. Geom. 7 (1998), 589–609.
  • A. Kumar, K3 surfaces associated with curves of genus two, Int. Math. Res. Not. (2008), 1073–7928.
  • ––––, Elliptic fibrations on a generic Jacobian Kummer surface, arXiv:1105.1715, J. Alg. Geom., accepted.
  • A. Mehran, Kummer surfaces associated to a $(1,2)$-polarized abelian surfaces, Nagoya Math. J. 202 (2011), 127–143.
  • D. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.
  • S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), 183–221.
  • I. Naruki, On metamorphosis of Kummer surfaces, Hokkaido Math. J. 20 (1991), 407–415.
  • V.V. Nikulin, Kummer surfaces, Izv. Akad. Nauk 39 (1975), 278–293; Math. USSR. Izv. 9 (1975), 261–275 (in English).
  • ––––, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv. 14 (1980), 103–167.
  • ––––, Finite groups of automorphisms of Kählerian surfaces of type K3, Trudy Mosk. Mat. Ob. 38 (1979), 75–137; Trans. Moscow Math. Soc. 38 (1980), 71–135 (in English).
  • K. Oguiso, On Jacobian fibrations on the Kummer surfaces of the product of non-isogenous elliptic curve, J. Math. Soc. Japan 41 (1989), 651–680.
  • H. Önisper and S. Sertöz, Generalized Shioda-Inose structures on K3 surfaces, Manuscr. Math. 98 (1999), 491–495.
  • I. Piatetski-Shapiro and I.R. Shafarevich, Torelli's theorem for algebraic surfaces of type K3, Izv. Akad. Nauk. 35 (1971), 530–572; Math. USSR, Izv. 5 (1971), 547–587 (in English).
  • B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.
  • M. Schütt, Sandwich theorems for Shioda-Inose structures, Izv. Mat. 77 (2013), 211–222.
  • I. Shimada, On elliptic K3 surfaces, Michigan Math. J. 47 (2000), 423–446.
  • T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59.
  • ––––, Some results on unirationality of algebraic surfaces, Math. Ann. 230 (1977), 153–168.
  • G. Xiao, Galois covers between K3 surfaces, Ann. Inst. Fourier 46 (1996), 73–88.