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Transactions of the American Mathematical Society

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On positivity of line bundles on Enriques surfaces

Author: Tomasz Szemberg
Journal: Trans. Amer. Math. Soc. 353 (2001), 4963-4972
MSC (2000): Primary 14J28; Secondary 14C20, 14E25
Published electronically: July 30, 2001
MathSciNet review: 1852089
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Abstract | References | Similar Articles | Additional Information


We study linear systems on Enriques surfaces. We prove rationality of Seshadri constants of ample line bundles on Enriques surfaces and provide lower bounds on these numbers. We show the nonexistence of $k$-very ample line bundles on Enriques surfaces of degree $4k+4$ for $k\geq 1$, thus answering an old question of Ballico and Sommese.

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  • 1. Ballico, E., Sommese, A.: Projective surfaces with $k$-very ample line bundles of degree $\leq 4k+4$, Nagoya Math. J. 136, 57-79 (1994) MR 96d:14005
  • 2. Barth, W., Peters, C., van de Ven, A.: Compact complex surfaces. Ergeb. Math. Grenzgeb. (3) 4, Springer-Verlag, 1984. MR 86c:32026
  • 3. Bauer, Th.: Seshadri constants on algebraic surfaces. Habilitationsschrift, Univ. Erlangen-Nürnberg, 1998. Math. Ann. 313, 547-583 (1999) MR 2000d:14006
  • 4. Beltrametti, M.C., Sommese, A.J.: Zero cycles and $k$-th order embeddings. Projective surfaces and their classification, Symp. Math., INDAM 32, 33-48, Academic Press 1991 MR 95d:14005
  • 5. Cossec, F.: Projective models of Enriques surfaces, Math. Ann. 265, 283-334 (1983) MR 86d:14035
  • 6. Cossec, F., Dolgachev I.: Enriques surfaces I, Progress in Mathematics 76, Birkhäuser 1989 MR 90h:14052
  • 7. 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
  • 8. Langer, A.: A note on $k$-jet ampleness on surfaces, in Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 273-282, Contemp. Math., 241, Amer. Math. Soc., Providence, RI, 1999 MR 2000j:14013
  • 9. Lazarsfeld, R.: Lectures on linear series. Park City/IAS Mathematics series vol. 3, 161-219 (1997) MR 98h:14008
  • 10. -, Multiplier Ideals for Algebraic Geometers, rlaz/
  • 11. Oguiso, K.: Seshadri constants for a family of surfaces, preprint, Essen 2000, forscher/alggeo.html
  • 12. Okonek, Ch.: Flächen vom Grad $8$ in $\mathbb P^4$, Math. Z. 191, 207-223 (1986) MR 87m:14058
  • 13. Reider, I.: Vector bundles of rank $2$ and linear systems on algebraic surfaces. Ann. Math. 127, 309-316 (1988) MR 89e:14038
  • 14. Terakawa, H.: Higher order embeddings of algebraic surfaces of Kodaira dimension zero, Math. Z. 229, 417-433 (1998) MR 99m:14013
  • 15. Verra, A.: On Enriques surfaces as a fourfold cover of $\mathbb P^2$, Math. Ann. 266, 241-250 (1983) MR 85b:14051

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Additional Information

Tomasz Szemberg
Affiliation: Institute of Mathematics, Reymonta 4, PL-30-059 Kraków, Poland
Address at time of publication: Department of Mathematics, University of Essen, FB6 Mathematik, D-45117 Essen, Germany

Keywords: Enriques surfaces, Seshadri constants, generation of jets, $k$-very ampleness
Received by editor(s): June 1, 2000
Published electronically: July 30, 2001
Additional Notes: Partially supported by KBN grant 2 P03A 00816
Article copyright: © Copyright 2001 American Mathematical Society

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