On positivity of line bundles on Enriques surfaces

Szemberg, Tomasz

doi:10.1090/S0002-9947-01-02788-X

On positivity of line bundles on Enriques surfaces
HTML articles powered by AMS MathViewer

by Tomasz Szemberg PDF

Trans. Amer. Math. Soc. 353 (2001), 4963-4972 Request permission

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

Edoardo Ballico and Andrew J. Sommese, Projective surfaces with $k$-very ample line bundles of degree $\leq 4k+4$, Nagoya Math. J. 136 (1994), 57–79. MR 1309381, DOI 10.1017/S002776300002496X
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
Thomas Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (1999), no. 3, 547–583. MR 1678549, DOI 10.1007/s002080050272
M. Beltrametti and A. J. Sommese, Zero cycles and $k$th order embeddings of smooth projective surfaces, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 33–48. With an appendix by Lothar Göttsche. MR 1273371
François R. Cossec, Projective models of Enriques surfaces, Math. Ann. 265 (1983), no. 3, 283–334. MR 721398, DOI 10.1007/BF01456021
François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969, DOI 10.1007/978-1-4612-3696-2
Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87–104. MR 1178721, DOI 10.1007/BFb0094512
Adrian Langer, A note on $k$-jet ampleness on surfaces, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 273–282. MR 1720870, DOI 10.1090/conm/241/03639
Robert Lazarsfeld, Lectures on linear series, Complex algebraic geometry (Park City, UT, 1993) IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI, 1997, pp. 161–219. With the assistance of Guillermo Fernández del Busto. MR 1442523, DOI 10.1090/pcms/003/03
—, Multiplier Ideals for Algebraic Geometers, http://www.math.lsa.umich.edu/~rlaz/MIAG.ps
Oguiso, K.: Seshadri constants for a family of surfaces, preprint, Essen 2000, http://www.expmath.uniessen.de/~forscher/alggeo.html
Christian Okonek, Flächen vom Grad $8$ im $\textbf {P}^4$, Math. Z. 191 (1986), no. 2, 207–223 (German). MR 818665, DOI 10.1007/BF01164025
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
Hiroyuki Terakawa, Higher order embeddings of algebraic surfaces of Kodaira dimension zero, Math. Z. 229 (1998), no. 3, 417–433. MR 1658577, DOI 10.1007/PL00004664
Alessandro Verra, On Enriques surface as a fourfold cover of $\textbf {P}^{2}$, Math. Ann. 266 (1983), no. 2, 241–250. MR 724741, DOI 10.1007/BF01458446