Nagoya Mathematical Journal

Every curve of genus not greater than eight lies on a $K3$ surface

Manabu Ide

Full-text: Open access


Let $C$ be a smooth irreducible complete curve of genus $g \geq 2$ 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 $2 \leq g \leq 8$ have ample $K3$ extensions. We use Bertini type lemmas and double coverings to construct ample $K3$ extensions.

Article information

Nagoya Math. J., Volume 190 (2008), 183-197.

First available in Project Euclid: 23 June 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H45: Special curves and curves of low genus 14C20: Divisors, linear systems, invertible sheaves 14J28: $K3$ surfaces and Enriques surfaces


Ide, Manabu. Every curve of genus not greater than eight lies on a $K3$ surface. Nagoya Math. J. 190 (2008), 183--197.

Export citation


  • E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of Algebraic Curves Vol. I., Springer-Verlag, New York, 1985.
  • D. Eisenbud, H. Lange, G. Martens and F.-O. Schreyer, The Clifford dimension of a projective curve, Compositio. Math., 72 (1989), 173--204.
  • P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, Inc., New York, 1978.
  • S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus $11$, Lecture Notes in Math. 1016, Springer-Verlag, 1983, pp. 334--353.
  • S. Mukai, Curves and symmetric spaces, Proc. Japan Acad., 68 (1992), 7--10.
  • S. Mukai, Curves and Grassmannians, Algebraic Geometry and Related Topics, Inchoen, Korea, 1992, International Press, Boston, 1993, pp. 19--40.
  • S. Mukai, Curves and symmetric spaces, I, Amer. J. Math., 117 (1995), 1627--1644.
  • S. Mukai, Curves, $K3$ surfaces and Fano $3$-folds of genus $\leq 10$, Algebraic Geometry and Commutative Algebra, Vol. 1, Kinokuniya, Tokyo, 1988, pp. 357--377.
  • S. Mukai and M. Ide, Canonical curves of genus eight, Proc. Japan Acad., 77 (2003), 59--64.
  • M. Reid, Chapters on algebraic surfaces, Complex Algebraic Geometry (Park City, UT, 1993), IAS/Park City Math. Ser., 3, Amer. Math. Soc., Providence, RI, 1997, pp. 3--159.
  • F.-O. Schreyer, Syzygies of canonical curves and special linear series, Math. Ann., 275 (1986), 105--137.
  • J. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J., 55 (1987), 843--871.