A Note on Seshadri constants on general <i>K</i>3 surfaces

Knutsen, Andreas Leopold

doi:10.1016/j.crma.2008.09.008

Algebraic Geometry

A Note on Seshadri constants on general K3 surfaces
[Une Note sur les constantes de Seshadri sur surfaces K3 générales]

Présenté par : Jean-Pierre Demailly

Knutsen, Andreas Leopold ¹

¹ Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway

Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1079-1081.

Résumé
Abstract

Nous démontrons une borne inférieure sur la constante de Seshadri $ε (L)$ sur un surface K3 telle que $Pic S ≃ Z [L]$ . En particulier, nous obténons que $ε (L) = α$ si $L^{2} = α^{2}$ pour un nombre entier α.

We prove a lower bound on the Seshadri constant $ε (L)$ on a K3 surface S with $Pic S ≃ Z [L]$ . In particular, we obtain that $ε (L) = α$ if $L^{2} = α^{2}$ for an integer α.

Reçu le : 2007-03-28
Accepté le : 2008-09-04
Publié le : 2008-09-23

DOI : 10.1016/j.crma.2008.09.008

Affiliations des auteurs :

Knutsen, Andreas Leopold ¹

¹ Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway

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Knutsen, Andreas Leopold. A Note on Seshadri constants on general K3 surfaces. Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1079-1081. doi : 10.1016/j.crma.2008.09.008. http://www.numdam.org/articles/10.1016/j.crma.2008.09.008/

Bibliographie
Cité par

[1] Bauer, Th. Seshadri constants on algebraic surfaces, Math. Ann., Volume 313 (1999), pp. 547-583

[2] Bauer, Th. Seshadri constants of quartic surfaces, Math. Ann., Volume 309 (1999), pp. 475-481

[3] Bauer, Th.; Di Rocco, S.; Szemberg, T. Generation of jets on K3 surfaces, J. Pure Appl. Algebra, Volume 146 (2000), pp. 17-27

[4] Chen, X. A simple proof that rational curves on K3 are nodal, Math. Ann., Volume 324 (2002), pp. 71-104

[5] Chen, X. Rational curves on K3 surfaces, J. Algebraic Geom., Volume 8 (1999), pp. 245-278

[6] Demailly, J.P. Singular Hermitian metrics on positive line bundles, Bayreuth, 1990 (Lecture Notes in Math.), Volume vol. 1507, Springer, Berlin (1992), pp. 87-104

[7] Kodaira, K. On the structure of compact complex analytic surfaces. I, Amer. J. Math., Volume 86 (1964), pp. 751-798

[8] Lazarsfeld, R. Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 48, Springer-Verlag, Berlin, 2004

[9] Oguiso, K. Seshadri constants in a family of surfaces, Math. Ann., Volume 323 (2002), pp. 625-631

[10] Steffens, A. Remarks on Seshadri constants, Math. Z., Volume 227 (1998), pp. 505-510

[11] Szemberg, T. On positivity of line bundles on Enriques surfaces, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 4963-4972

Cité par Sources :

^⁎ Research supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and carried out at Università di Roma Tre, Rome, Italy.