Donagi–Morrison’s examples on 2-elementary K3 surfaces

Abstract

Let X be an algebraic K3 surface, and let L be a base point free and big line bundle on X. If X admits a map of degree 2 to the projective plane branched over a smooth sextic and L is the pullback of the line bundle \({\mathcal{O}_{\mathbb{P}^{2}}(3),}\) then the gonality of the smooth curves of the complete linear system |L| is not constant. The polarized K3 surface (X, L) consisting of the K3 surface X and the line bundle L is called Donagi–Morrison’s example. In this paper, we give a necessary and sufficient condition for the polarized K3 surface (X, L) consisting of a 2-elementary K3 surface X and an ample line bundle L to be Donagi–Morrison’s example.