Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Hyperelliptic genus 4 curves on abelian surfaces

Authors: Paweł Borówka and G. K. Sankaran
Journal: Proc. Amer. Math. Soc. 145 (2017), 5023-5034
MSC (2010): Primary 14K10; Secondary 14H42
Published electronically: August 31, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study smooth curves on abelian surfaces, especially for genus $ 4$, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus $ 4$ hyperelliptic curve on a general $ (1,3)$-polarised abelian surface. We investigate these curves and show that their Jacobians contain a surface and its dual as complementary abelian subvarieties.

References [Enhancements On Off] (What's this?)

  • [AM] A. Andreotti and A. L. Mayer, On period relations for abelian integrals on algebraic curves, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 189-238. MR 0220740
  • [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
  • [B] Wolf Barth, Abelian surfaces with $ (1,2)$-polarization, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 41-84. MR 946234
  • [BN] W. Barth and I. Nieto, Abelian surfaces of type $ (1,3)$ and quartic surfaces with $ 16$ skew lines, J. Algebraic Geom. 3 (1994), no. 2, 173-222. MR 1257320
  • [BL] Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673
  • [BL2] Ch. Birkenhake and H. Lange, Moduli spaces of abelian surfaces with isogeny, Geometry and analysis (Bombay, 1992) Tata Inst. Fund. Res., Bombay, 1995, pp. 225-243. MR 1351509
  • [Bo] Paweł Borówka, Non-simple principally polarised abelian varieties, Ann. Mat. Pura Appl. (4) 195 (2016), no. 5, 1531-1549. MR 3537961,
  • [BO] P. Borówka and A. Ortega, Hyperelliptic curves on $ (1, 4)$ polarised abelian surfaces, arXiv:1708.01270.
  • [BOPY] J. Bryan, G. Oberdieck, R. Pandharipande, and Q. Yin, Curve counting on abelian surfaces and threefolds, to appear in Algebraic Geometry.
  • [M] David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325-350. MR 0379510
  • [Ri] John Ries, The Prym variety for a cyclic unramified cover of a hyperelliptic Riemann surface, J. Reine Angew. Math. 340 (1983), 59-69. MR 691961,
  • [Ro] Simon C. F. Rose, Counting hyperelliptic curves on an Abelian surface with quasi-modular forms, Commun. Number Theory Phys. 8 (2014), no. 2, 243-293. MR 3271176,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14K10, 14H42

Retrieve articles in all journals with MSC (2010): 14K10, 14H42

Additional Information

Paweł Borówka
Affiliation: Institute of Mathematics, Jagiellonian University in Kraków, ul. prof Stanisława Łojasiewicza 6, 30-348 Kraków, Poland

G. K. Sankaran
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, England

Received by editor(s): July 21, 2016
Published electronically: August 31, 2017
Communicated by: Lev Borisov
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society