Smooth Curves on Projective $K3$ Surfaces - Knutsen, Andreas Leopold (MATHEMATICA SCANDINAVICA, 2002)
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 $P^{n+1}$ 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 $\mathcal O_C (k)$ to be non-special, for any integer $k ≥ 1$.