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

Abstract

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.