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⊗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≥ 2 , g>2d-3+(r-1)^2 , d≥ 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 π :C⟶ C_0 of a smooth curve C_0⊂ℙ^2 of degree k≥ 4 branched at distinct 6k points on C_0 , then, by using the aforementioned result, we can also show that there exists a 2:1 morphism π̃:X⟶ℙ^2 such that π̃|_C=π .