Lifts of line bundles on curves on K3 surfaces

Let X be a K3 surface, let C be a smooth curve of genus g on X, and let A be a line bundle of degree d on C. Then a line bundle M on X with \(M\otimes {\mathcal {O}}_C=A\) $M\otimes {\mathcal{O}}_C=A$ is called a lift of A. In this paper, we prove that if the dimension of the linear system |A| is \(r\ge 2\) $r\ge 2$ , \(g>2d-3+(r-1)^2\) $g>2d-3+{(r-1)}^2$ , \(d\ge 2r+4\) $d\ge 2r+4$ , and A computes the Clifford index of C, then there exists a base point free lift M of A such that the general member of |M| is a smooth curve of genus r. In particular, if |A| is a base point free net which defines a double covering \(\pi :C\longrightarrow C_0\) $\pi :C⟶C_0$ of a smooth curve \(C_0\subset {\mathbb {P}}^2\) $C_0\subset {\mathbb{P}}^2$ of degree \(k\ge 4\) $k\ge 4$ branched at distinct 6k points on \(C_0\) $C_0$ , then, by using the aforementioned result, we can also show that there exists a 2:1 morphism \({\tilde{\pi }}:X\longrightarrow {\mathbb {P}}^2\) $\tilde{\pi }:X⟶{\mathbb{P}}^2$ such that \({\tilde{\pi }}|_C=\pi \) $\tilde{\pi }{|}_C=\pi$ .