Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T04:05:55.852Z Has data issue: false hasContentIssue false

Every Curve of Genus not Greater Than Eight Lies on a K3 Surface

Published online by Cambridge University Press:  11 January 2016

Manabu Ide*
Affiliation:
Graduate School of Mathematics, Nagoya University, Furō-chō, Chikusa-ku, Nagoya 464-8602, Japan
*
Tokoha Gakuen University, 1-22-1, Sena, Aoi-ku, Shizuoka-shi, 420-0911, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let C be a smooth irreducible complete curve of genus g ≥ 2 over an algebraically closed field of characteristic 0. An ample K3 extension of C is a K3 surface with at worst rational double points which contains C in the smooth locus as an ample divisor.

In this paper, we prove that all smooth curve of genera. 2 ≤ g ≤ 8 have ample K3 extensions. We use Bertini type lemmas and double coverings to construct ample K3 extensions.

Keywords

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[ACGH] Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J., Geometry of Algebraic Curves Vol. I., Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
[ELMS] Eisenbud, D., Lange, H., Martens, G. and Schreyer, F.-O., The Clifford dimension of a projective curve, Compositio. Math., 72 (1989), 173204.Google Scholar
[GH] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, John Wiley & Sons, Inc., New York, 1978.Google Scholar
[MM] Mori, S. and Mukai, S., The uniruledness of the moduli space of curves of genus 11, Lecture Notes in Math. 1016, Springer-Verlag, 1983, pp. 334353.Google Scholar
[M1] Mukai, S., Curves and symmetric spaces, Proc. Japan Acad., 68 (1992), 710.Google Scholar
[M2] Mukai, S., Curves and Grassmannians, Algebraic Geometry and Related Topics, Inchoen, Korea, 1992, International Press, Boston, 1993, pp. 1940.Google Scholar
[M3] Mukai, S., Curves and symmetric spaces, I, Amer. J. Math., 117 (1995), 16271644.CrossRefGoogle Scholar
[M4] Mukai, S., Curves, K3 surfaces and Fano 3-folds of genus ≤ 10, Algebraic Geometry and Commutative Algebra, Vol. 1, Kinokuniya, Tokyo, 1988, pp. 357377.CrossRefGoogle Scholar
[MI] Mukai, S. and Ide, M., Canonical curves of genus eight, Proc. Japan Acad., 77 (2003), 5964.Google Scholar
[R] Reid, M., Chapters on algebraic surfaces, Complex Algebraic Geometry (Park City, UT, 1993), IAS/Park City Math. Ser., 3, Amer. Math. Soc., Providence, RI, 1997, pp. 3159.Google Scholar
[S] Schreyer, F.-O., Syzygies of canonical curves and special linear series, Math. Ann., 275 (1986), 105137.Google Scholar
[W] Wahl, J., The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J., 55 (1987), 843871.Google Scholar