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