Advertisement

Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 193–209 | Cite as

Hyperelliptic curves on (1, 4)-polarised abelian surfaces

  • Paweł Borówka
  • Angela OrtegaEmail author
Article
  • 16 Downloads

Abstract

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We give a short proof for the known fact the only possibilities of genera of such curves are 2, 3, 4 and 5; then we focus on the genus five case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general (1, 4)-polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. Our proof of the existence of hyperelliptic curves on general (1, 4)-polarised abelian surfaces is different from that in the recent paper [10]. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every étale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of 2-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.

Mathematics Subject Classification

14H40 14H30 

Notes

Acknowledgements

We thank the referee for pointing out a couple of crucial references for this paper.

References

  1. 1.
    Andreotti, A., Mayer, A.L.: On period relations for abelian integrals on algebraic curves. Ann. Sc. Norm. Super. Pisa III Ser. 21(2), 189–238 (1967)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Grundlehren der Math. Wiss, vol. 1, p. 267. Springer-Verlag, New York (1984)Google Scholar
  3. 3.
    Birkenhake, Ch., Lange, H.: Moduli spaces of abelian surfaces with isogeny. Geometry and analysis (Bombay, 1992), pp. 225–243, Tata Inst. Fund. Res., Bombay (1995)Google Scholar
  4. 4.
    Birkenhake, Ch., Lange, H.: Complex Abelian Varieties, Grundlehren der Mathematischen Wissenschaften, 2nd edn, p. 302. Springer-Verlag, Berlin (2004)CrossRefGoogle Scholar
  5. 5.
    Birkenhake, Ch., Lange, H., van Straten, D.: Abelian surfaces of type (1,4). Math. Ann. 285, 625–646 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borówka, P., Sankaran, G.K.: Hyperelliptic genus 4 curves on abelian surfaces. Proc. Amer. Math. Soc. 145, 5023–5034 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bryan, J., Oberdieck, G., Pandharipande, R., Yin, Q.: Curve counting on abelian surfaces and threefolds. Algebr. Geom. 5(4), 398–463 (2018)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dolgachev, I.: Classical Algebraic Geometry. A modern view. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Farkas, H.: Unramified double coverings of hyperelliptic surfaces. J. Anal. Math. 30, 150–155 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Knutsen, A.L., Lelli-Chiesa, M., Mongardi, G.: Severi varieties and Brill-Noether theory on abelian surfaces, J. reine angew. Math,  https://doi.org/10.1515/crelle-2016-0029
  11. 11.
    Maclachlan, C.: Smooth coverings of hyperelliptic surfaces. Q. J. Math. Oxf. 22(1), 117–123 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mumford, D.: Prym varieties I. In: Ahlfors, L.V., Kra, I., Maskit, B., Nirenberg, L. (eds.) Contributions to Analysis, pp. 325–350. Academic Press, New York (1974)CrossRefGoogle Scholar
  13. 13.
    Recillas, S., Rodriguez, R.: Prym varieties and fourfolds covers, Publ. Preliminares Inst. Mat. Univ. Nac. Aut. Mexico, 686 (2001). arXiv:math/0303155
  14. 14.
    Reider, I.: Vector bundles of rank 2 and linear systems on algebraic surfaces. Ann. Math. 127, 309–316 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsJagiellonian University in KrakówKrakówPoland
  2. 2.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany

Personalised recommendations