Smooth curves on projective K3 surfaces

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In this paper we give for all n ≥ 2, d>0, g ≥ 0 necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in Pⁿ⁺¹ and C is a smooth (reduced and irreducible) curve of degree d and genus g on X. The surfaces constructed have Picard group of minimal rank possible (being either 1 or 2), and in each case we specify a set of generators. For n ≥ 4 we also determine when X can be chosen to be an intersection of quadrics (in all other cases X has to be an intersection of both quadrics and cubics). Finally, we give necessary and sufficient conditions for ØC (k) to be non-special, for any integer k ≥ 1.