Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-01-02788-X
Published electronically: July 30, 2001
MathSciNet review: 1852089
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

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.


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

  • 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, http://www.math.lsa.umich.edu/ rlaz/MIAG.ps
  • 11. Oguiso, K.: Seshadri constants for a family of surfaces, preprint, Essen 2000, http://www.expmath.uniessen.de/ 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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14J28, 14C20, 14E25

Retrieve articles in all journals with MSC (2000): 14J28, 14C20, 14E25


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
Email: szemberg@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-01-02788-X
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

American Mathematical Society