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Nikulin Involutions and the CHL String

  • Adrian Clingher
  • Andreas MalmendierEmail author
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Abstract

We study certain even-eight curve configurations on a specific class of Jacobian elliptic K3 surfaces with lattice polarizations of rank ten. These configurations are associated with K3 double covers, some of which are elliptic but not Jacobian elliptic. Several non-generic cases corresponding to K3 surfaces of higher Picard rank are also discussed. Finally, the results and the construction in question are interpreted in the context of the string dualities linked with the eight-dimensional CHL string.

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Notes

Acknowledgements

We thank the referees for many helpful comments.

References

  1. 1.
    Barth, W.: Even Sets of Eight Rational Curves on a K3-Surface, Complex Geometry (Göttingen, 2000), pp. 1–25. Springer, Berlin (2002)Google Scholar
  2. 2.
    Bershadsky M., Tony P., Sadov V.: F-theory with quantized fluxes. Adv. Theor. Math. Phys. 3(3), 727–773 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Braun V., Tony D.R.: F-theory on genus-one fibrations. J. High Energy Phys. 132(8), front matter+45 (2014)MathSciNetGoogle Scholar
  4. 4.
    Clingher A., Doran C.F.: Note on a geometric isogeny of K3 surfaces. Int. Math. Res. Not. IMRN 2011(16), 3657–3687 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clingher, A., Malmendier, A.: On the geometry of (1,2)-polarized Kummer surfaces, arXiv:1704.04884 [math.AG].
  6. 6.
    Comparin P., Garbagnati A.: Van Geemen–Sarti involutions and elliptic fibrations on K3 surfaces double cover of \({{\mathbb{P}}^2}\). J. Math. Soc. Jpn. 66(2), 479–522 (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cossec, F.R., Dolgachev, I.V.: Enriques Surfaces. I, Progress in Mathematics, vol. 76. Birkhäuser Boston, Inc., Boston, MA (1989)Google Scholar
  8. 8.
    Dolgachev I.V.: Mirror symmetry for lattice polarized K3 surfaces. J. Math. Sci. 81(3), 2599–2630 (1996) Algebraic geometry, 4MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dolgachev, I.V.: A brief introduction to Enriques surfaces, Development of moduli theory—Kyoto 2013, Advanced Studies in Pure Mathematics, vol. 69, pp. 1–32. Mathematics Society Japan, Tokyo (2016)Google Scholar
  10. 10.
    Friedman R., Morgan J.W., Witten E.: Vector bundles over elliptic fibrations. J. Algebraic Geom. 8(2), 279–401 (1999)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hoyt, W.L., Schwartz, C.F.: Yoshida surfaces with Picard number \({\rho\geq 17}\). In: Proceedings on Moonshine and Related Topics (Montréal, QC, 1999), CRM Proceedings Lecture Notes, vol. 30, pp. 71–78. American Mathematical Society, Providence, RI (2001)Google Scholar
  12. 12.
    Hudson, R.W.H.T.: Kummer’s Quartic Surface. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1990), With a foreword by W. Barth, Revised reprint of the 1905 original.Google Scholar
  13. 13.
    Inose H.: On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P 3. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(3), 545–560 (1976)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kloosterman R.: Classification of all Jacobian elliptic fibrations on certain K3 surfaces. J. Math. Soc. Jpn. 58(3), 665–680 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kodaira K.: On compact analytic surfaces. II. Ann. Math. (2) 77, 563–626 (1963)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kodaira K.: On compact analytic surfaces. III. Ann. Math. (2) 78, 1–40 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Koike K., Shiga H., Takayama N., Tsutsui T.: Study on the family of K3 surfaces induced from the lattice \({(D_4)^3\oplus\langle-2\rangle\oplus\langle 2\rangle}\). Int. J. Math. 12(9), 1049–1085 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kumar A.: Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. 23(4), 599–667 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lerche W., Schweigert C., Minasian R., Theisen S.: A note on the geometry of CHL heterotic strings. Phys. Lett. B 424(1-2), 53–59 (1998)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Malmendier A., Morrison D.R.: K3 surfaces, modular forms, and non-geometric heterotic compactifications. Lett. Math. Phys. 105(8), 1085–1118 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mehran A.: Double covers of Kummer surfaces. Manuscr. Math. 123(2), 205–235 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mehran A.: Kummer surfaces associated to (1,2)-polarized abelian surfaces. Nagoya Math. J. 202, 127–143 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Morrison D.R.: On K3 surfaces with large Picard number. Invent. Math. 75(1), 105–121 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Morrison D.R., Vafa C.: Compactifications of F-theory on Calabi–Yau threefolds. I. Nucl. Phys. B 473(1-2), 74–92 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Morrison D.R., Vafa C.: Compactifications of F-theory on Calabi–Yau threefolds. II. Nucl. Phys. B 476(3), 437–469 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Narain K.S.: New heterotic string theories in uncompactified dimensions < 10. Phys. Lett. B 169(1), 41–46 (1986)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Nikulin V.V.: Finite groups of automorphisms of Kählerian K3 surfaces. Trudy Moskov. Mat. Obshch. 38, 75–137 (1979)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Oguiso K., Shioda T.: The Mordell–Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40(1), 83–99 (1991)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Shimada I.: On elliptic K3 surfaces. Mich. Math. J. 47(3), 423–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    van Geemen B., Sarti A.: Nikulin involutions on K3 surfaces. Math. Z. 255(4), 731–753 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Witten, E.: Toroidal compactification without vector structure. J. High Energy Phys. no. 2, Paper 6, 43 (1998)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Missouri – St. LouisSt. LouisUSA
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

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