## Nagoya Mathematical Journal

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

Manabu Ide

#### Abstract

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

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

Dates
First available in Project Euclid: 23 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1214229082

Mathematical Reviews number (MathSciNet)
MR2423833

Zentralblatt MATH identifier
1140.14305

#### Citation

Ide, Manabu. Every curve of genus not greater than eight lies on a $K3$ surface. Nagoya Math. J. 190 (2008), 183--197. https://projecteuclid.org/euclid.nmj/1214229082

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