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Transactions of the American Mathematical Society

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Primitive higher order embeddings
of abelian surfaces

Authors: Th. Bauer and T. Szemberg
Journal: Trans. Amer. Math. Soc. 349 (1997), 1675-1683
MSC (1991): Primary 14E25; Secondary 14C20
MathSciNet review: 1376538
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Abstract: In recent years several concepts of higher order embeddings have been studied: $k$-spannedness, $k$-very ampleness and $k$-jet ampleness. In the present note we consider primitive line bundles on abelian surfaces and give numerical criteria which allow to check whether a given ample line bundle satisfies these properties.

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  • 1. Ballico, E., Sommese, A.J.: Projective surfaces with $k$-very ample line bundles of degree $\le 4k+4$. Nagoya Math. J. 136, 57-79 (1994) MR 96d:14005
  • 2. Barth, W.: Abelian Surfaces with $(1,2)$-Polarisation. Algebraic Geometry (Sendai, 1985), Advanced Studies in Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 41-84. MR 89i:14038
  • 3. Bauer, Th.: Quartic surfaces with 16 skew conics. J. Reine Angew. Math. 464, 207-217 (1995) MR 96j;14024
  • 4. Bauer, Th., Szemberg, T.: Higher order embeddings of abelian varieties. Math. Z. (to appear)
  • 5. Beltrametti, M.C., Sommese, A.J.: Zero cycles and $k$-th order embeddings of smooth projective surfaces. Projective Surfaces and Their Classification, Symp. Math., INDAM, vol. 32, Academic Press, New York, 1988, pp. 33-48. MR 95d:14005
  • 6. Beltrametti, M.C., Sommese, A. J.: On $k$-jet ampleness. Complex Analysis and Geometry, edited by V. Ancona and A. Silva, Plenum Press, New York, 1993, pp. 355-376. MR 94g:14006
  • 7. Beltrametti, M.C., Sommese, A.J.: The adjunction theory of complex projective varieties. De Gruyter Expositions in Math. 16, de Gruyter, Berlin, 1995. MR 96f:14004
  • 8. Birkenhake, Ch.: Tensor products of ample line bundles on abelian varieties. Manuscripta Math. 84, 21-28 (1994) MR 95i:14040
  • 9. Birkenhake, Ch., Lange, H.: A family of abelian surfaces and curves of genus four. Manuscripta Math. 85, 393-407 (1994) MR 95k:14064
  • 10. Birkenhake, Ch., Lange, H., van Straten, D.: Abelian surfaces of type $(1,4)$. Math. Ann. 285, 625-646 (1989) MR 91b:14042
  • 11. Debarre, O., Hulek, K., Spandaw, J.: Very ample linear systems on abelian varieties. Math. Ann. 300, 181-202 (1994) MR 95k:14065
  • 12. Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. Complex Algebraic Varieties (Bayreuth, 1990), Lect. Notes Math. 1507, Springer-Verlag, 1992, pp. 87-104. MR 93g:32044
  • 13. Demailly, J.-P.: $L^2$ vanishing theorems for positive line bundles and adjunction theory. To appear.
  • 14. Ein, L., Küchle, O., Lazarsfeld, R.: Local positivity of ample line bundles. J. Diff. Geom. 42, 193-219 (1995) MR 96m:14007
  • 15. Ein, L., Lazarsfeld, R.: Global generation of pluricanonical and adjoint linear series on smooth projective threefolds. J. Amer. Math. Soc. 6, 875-903 (1993) MR 94c:14016
  • 16. Ein, L., Lazarsfeld, R.: Seshadri constants on smooth surfaces. Astérisque 218, 177-186 (1993) MR 95f:14031
  • 17. Hulek, K., Lange, H.: Examples of abelian surfaces in ${\mathbb {P}} _4$. J. Reine Angew. Math. 363, 200-216 (1985) MR 87g:14038
  • 18. Lange, H., Birkenhake, Ch.: Complex Abelian Varieties. Springer-Verlag, 1992. MR 94j:14001
  • 19. Lazarsfeld, R.: Lectures on linear series. Park City / IAS Mathematics series vol. 3, 1-56 (1993)
  • 20. Steffens, A.: Remarks on Seshadri constants. Preprint
  • 21. Zak, F.: Linear systems of hyperplane sections on varieties of low codimension. Functional Anal. Appl. 19, 165-173 (1985) MR 87d:14040 (of Russian original)

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Additional Information

Th. Bauer
Affiliation: Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße $1\frac12$, D-91054 Erlangen, Germany

T. Szemberg
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, PL-30-059 Kraków, Poland

Received by editor(s): December 1, 1995
Additional Notes: The first author was supported by DFG contract Ba 423/7-1
The second author was partially supported by KBN grant P03A-061-08. The final version of this paper was written during the second author’s stay in Erlangen, which was made possible by Europroj support.
Article copyright: © Copyright 1997 American Mathematical Society

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