$K3$ curves with index $k>1$

Let $mathcal{KC}_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface and $Csubset S$ a genus $g$ curve with divisibility $k$ in $mathrm{Pic}(S)$. In this article we study the forgetful map $c_g^k:(S,C) mapsto C$ from $mathcal{KC}_g ^k$ to $mathcal{M}_g$ for $k>1$. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when $S$ is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending $C$ in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $c_g^k$ dominates the locus in $mathcal{M}_g$ of $k$-spin curves with the appropriate number of independent sections. We are able to do this only when $S$ is a complete intersection, and obtain in these cases some classification results for spin curves.