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 > 2 d - 3 + (r - 1) 2 , d ≥ 2 r + 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 π ~ : X ⟶ P 2 such that π ~ | C = π . [ABSTRACT FROM AUTHOR]