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)

 

 

A note on the normal generation of ample line bundles on an abelian surface


Author: Akira Ohbuchi
Journal: Proc. Amer. Math. Soc. 117 (1993), 275-277
MSC: Primary 14J25; Secondary 14K05
MathSciNet review: 1106182
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L$ be an ample line bundle on an abelian surface $ A$. We prove that the four conditions: (1) $ L$ is base point free, (2) $ L$ is fixed component free, (3) $ {L^{ \otimes 2}}$ is very ample, (4) $ {L^{ \otimes 2}}$ is normally generated, are equivalent if $ ({L^2}) > 4$. Moreover we prove that $ {L^{ \otimes 2}}$ is not normally generated if $ ({L^2}) = 4$.


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

  • [1] Shoji Koizumi, Theta relations and projective normality of Abelian varieties, Amer. J. Math. 98 (1976), no. 4, 865–889. MR 0480543
  • [2] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985
  • [3] Akira Ohbuchi, Some remarks on ample line bundles on abelian varieties, Manuscripta Math. 57 (1987), no. 2, 225–238. MR 871633, 10.1007/BF02218082
  • [4] Akira Ohbuchi, A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), no. 4, 119–120. MR 966402
  • [5] Ryuji Sasaki, Theta relations and their applications in abstract geometry, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1977), no. 366-382, 290–312. MR 0476767
  • [6] Tsutomu Sekiguchi, On projective normality of Abelian varieties, J. Math. Soc. Japan 28 (1976), no. 2, 307–322. MR 0401784
  • [7] Tsutomu Sekiguchi, On projective normality of Abelian varieties. II, J. Math. Soc. Japan 29 (1977), no. 4, 709–727. MR 0457457
  • [8] Tsutomu Sekiguchi, On the cubics defining abelian varieties, J. Math. Soc. Japan 30 (1978), no. 4, 703–721. MR 513079, 10.2969/jmsj/03040703
  • [9] Oscar Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, no. 4, The Mathematical Society of Japan, Tokyo, 1958. MR 0097403

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14J25, 14K05

Retrieve articles in all journals with MSC: 14J25, 14K05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1106182-7
Article copyright: © Copyright 1993 American Mathematical Society